{"id":95817,"date":"2026-08-26T23:08:38","date_gmt":"2026-08-26T15:08:38","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/95817.html"},"modified":"2026-08-26T23:08:38","modified_gmt":"2026-08-26T15:08:38","slug":"%e4%b8%8d%e5%8a%a8%e5%9f%9f%e5%92%8c%e4%bc%bd%e7%bd%97%e7%93%a6%e5%af%b9%e5%ba%94","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/95817.html","title":{"rendered":"\u4e0d\u52a8\u57df\u548c\u4f3d\u7f57\u74e6\u5bf9\u5e94"},"content":{"rendered":"

\u53c2\u8003<\/h2>\n

\u5206\u5706\u65b9\u7a0bx^17-1=0\u7684\u89e3\u6cd5.csdn<\/p>\n

\u57df\u4e0a\u7684\u81ea\u540c\u6784<\/h2>\n

\u8bbe <\/p>\n

L<\/p>\n

L<\/p>\n

<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span> \u662f\u4e00\u4e2a\u57df\u3002<\/p>\n

\u5982\u679c\u6620\u5c04：<\/p>\n

<\/p>\n

\u03c3<\/p>\n

:<\/p>\n

L<\/p>\n

\u27f6<\/p>\n

L<\/p>\n

\\\\sigma:L\\\\longrightarrow L <\/p>\n

<\/span><\/span>\u03c3<\/span><\/span>:<\/span><\/span><\/span><\/span>L<\/span><\/span>\u27f6<\/span><\/span><\/span><\/span>L<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6ee1\u8db3：<\/p>\n

<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

a<\/p>\n

+<\/p>\n

b<\/p>\n

)<\/p>\n

=<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

a<\/p>\n

)<\/p>\n

+<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

b<\/p>\n

)<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

a<\/p>\n

b<\/p>\n

)<\/p>\n

=<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

a<\/p>\n

)<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

b<\/p>\n

)<\/p>\n

\\\\boxed{ \\\\begin{aligned} \\\\sigma(a+b)&=\\\\sigma(a)+\\\\sigma(b)\\\\\\\\ \\\\sigma(ab)&=\\\\sigma(a)\\\\sigma(b)\\\\\\\\ \\\\end{aligned} }<\/p>\n

<\/span><\/span><\/span><\/span>\u03c3<\/span>(<\/span>a<\/span><\/span>+<\/span><\/span>b<\/span>)<\/span><\/span><\/span><\/span>\u03c3<\/span>(<\/span>ab<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u03c3<\/span>(<\/span>a<\/span>)<\/span><\/span>+<\/span><\/span>\u03c3<\/span>(<\/span>b<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u03c3<\/span>(<\/span>a<\/span>)<\/span>\u03c3<\/span>(<\/span>b<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5e76\u4e14 <\/p>\n

\u03c3<\/p>\n

\\\\sigma<\/p>\n

<\/span><\/span>\u03c3<\/span><\/span><\/span><\/span><\/span> \u662f\u53cc\u5c04，\u5219\u79f0 <\/p>\n

\u03c3<\/p>\n

\\\\sigma<\/p>\n

<\/span><\/span>\u03c3<\/span><\/span><\/span><\/span><\/span> \u662f <\/p>\n

L<\/p>\n

L<\/p>\n

<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span> \u7684\u4e00\u4e2a\u81ea\u540c\u6784\u3002<\/p>\n

\u8bb0\u6240\u6709\u81ea\u540c\u6784\u7ec4\u6210\u7684\u7fa4\u4e3a：<\/p>\n

<\/p>\n

Aut<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

)<\/p>\n

=<\/p>\n

{<\/p>\n

\u03c3<\/p>\n

:<\/p>\n

L<\/p>\n

\u2192<\/p>\n

L<\/p>\n

\u2223<\/p>\n

\u03c3<\/p>\n

\u00a0\u662f\u00a0<\/p>\n

L<\/p>\n

\u00a0\u7684\u81ea\u540c\u6784<\/p>\n

}<\/p>\n

\\\\boxed{\\\\operatorname{Aut}(L) =\\\\{\\\\sigma:L\\\\to L\\\\mid \\\\sigma\\\\text{ \u662f }L\\\\text{ \u7684\u81ea\u540c\u6784} \\\\} } <\/p>\n

<\/span><\/span><\/span>Aut<\/span><\/span>(<\/span>L<\/span>)<\/span><\/span>=<\/span><\/span>{<\/span>\u03c3<\/span><\/span>:<\/span><\/span>L<\/span><\/span>\u2192<\/span><\/span>L<\/span><\/span>\u2223<\/span><\/span>\u03c3<\/span>\u00a0<\/span>\u662f<\/span>\u00a0<\/span><\/span>L<\/span>\u00a0<\/span>\u7684\u81ea\u540c\u6784<\/span><\/span>}<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d\u7fa4\u8fd0\u7b97\u4e3a\u6620\u5c04\u7684\u590d\u5408\u3002<\/p>\n

\n

\u81ea\u540c\u6784\u4fdd\u6301\u57df\u8fd0\u7b97<\/h3>\n

\u5bf9\u4e8e\u4efb\u610f <\/p>\n

a<\/p>\n

,<\/p>\n

b<\/p>\n

\u2208<\/p>\n

L<\/p>\n

a,b\\\\in L<\/p>\n

<\/span><\/span>a<\/span>,<\/span><\/span>b<\/span><\/span>\u2208<\/span><\/span><\/span><\/span>L<\/span><\/span><\/span><\/span><\/span>，\u6709：<\/p>\n

<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

a<\/p>\n

\u2212<\/p>\n

b<\/p>\n

)<\/p>\n

=<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

a<\/p>\n

)<\/p>\n

\u2212<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

b<\/p>\n

)<\/p>\n

\\\\boxed{ \\\\sigma(a-b)=\\\\sigma(a)-\\\\sigma(b) } <\/p>\n

<\/span><\/span><\/span>\u03c3<\/span>(<\/span>a<\/span><\/span>\u2212<\/span><\/span>b<\/span>)<\/span><\/span>=<\/span><\/span>\u03c3<\/span>(<\/span>a<\/span>)<\/span><\/span>\u2212<\/span><\/span>\u03c3<\/span>(<\/span>b<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

a<\/p>\n

b<\/p>\n

)<\/p>\n

=<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

a<\/p>\n

)<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

b<\/p>\n

)<\/p>\n

\\\\boxed{ \\\\sigma(ab)=\\\\sigma(a)\\\\sigma(b) } <\/p>\n

<\/span><\/span><\/span>\u03c3<\/span>(<\/span>ab<\/span>)<\/span><\/span>=<\/span><\/span>\u03c3<\/span>(<\/span>a<\/span>)<\/span>\u03c3<\/span>(<\/span>b<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5f53 <\/p>\n

b<\/p>\n

\u2260<\/p>\n

0<\/p>\n

b\\\\neq0<\/p>\n

<\/span><\/span>b<\/span><\/span><\/span>\ue020<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span> \u65f6：<\/p>\n

<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

a<\/p>\n

b<\/p>\n

)<\/p>\n

=<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

a<\/p>\n

)<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

b<\/p>\n

)<\/p>\n

\\\\boxed{\\\\sigma\\\\left(\\\\frac ab\\\\right) =\\\\frac{\\\\sigma(a)}{\\\\sigma(b)} } <\/p>\n

<\/span><\/span><\/span>\u03c3<\/span><\/span>(<\/span><\/span><\/span><\/span>b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c3<\/span>(<\/span>b<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c3<\/span>(<\/span>a<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u6b64，\u81ea\u540c\u6784\u4fdd\u6301\u57df\u4e2d\u7684\u52a0\u3001\u51cf\u3001\u4e58\u3001\u9664\u8fd0\u7b97\u3002<\/p>\n

\n

\u81ea\u540c\u6784\u7684\u4f8b\u5b50<\/h3>\n

<\/p>\n

Q<\/p>\n

(<\/p>\n

2<\/p>\n

)<\/p>\n

\\\\mathbb Q(\\\\sqrt2)<\/p>\n

<\/span><\/span>Q<\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span> \u7684\u81ea\u540c\u6784<\/h4>\n

\u8bbe：<\/p>\n

<\/p>\n

L<\/p>\n

=<\/p>\n

Q<\/p>\n

(<\/p>\n

2<\/p>\n

)<\/p>\n

L=\\\\mathbb Q(\\\\sqrt2) <\/p>\n

<\/span><\/span>L<\/span><\/span>=<\/span><\/span><\/span><\/span>Q<\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5b9a\u4e49：<\/p>\n

<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

2<\/p>\n

)<\/p>\n

=<\/p>\n

\u2212<\/p>\n

2<\/p>\n

\\\\sigma(\\\\sqrt2)=-\\\\sqrt2 <\/p>\n

<\/span><\/span>\u03c3<\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5219\u5bf9\u4e8e\u4efb\u610f：<\/p>\n

<\/p>\n

a<\/p>\n

+<\/p>\n

b<\/p>\n

2<\/p>\n

\u2208<\/p>\n

L<\/p>\n

,<\/p>\n

a<\/p>\n

,<\/p>\n

b<\/p>\n

\u2208<\/p>\n

Q<\/p>\n

a+b\\\\sqrt2\\\\in L, \\\\qquad a,b\\\\in\\\\mathbb Q <\/p>\n

<\/span><\/span>a<\/span><\/span>+<\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2208<\/span><\/span><\/span><\/span>L<\/span>,<\/span><\/span><\/span>a<\/span>,<\/span><\/span>b<\/span><\/span>\u2208<\/span><\/span><\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6709：<\/p>\n

<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

a<\/p>\n

+<\/p>\n

b<\/p>\n

2<\/p>\n

)<\/p>\n

=<\/p>\n

a<\/p>\n

\u2212<\/p>\n

b<\/p>\n

2<\/p>\n

\\\\boxed{ \\\\sigma(a+b\\\\sqrt2)=a-b\\\\sqrt2 } <\/p>\n

<\/span><\/span><\/span>\u03c3<\/span>(<\/span>a<\/span><\/span>+<\/span><\/span>b<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>=<\/span><\/span>a<\/span><\/span>\u2212<\/span><\/span>b<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u6b64：<\/p>\n

<\/p>\n

Aut<\/p>\n

\u2061<\/p>\n

(<\/p>\n

Q<\/p>\n

(<\/p>\n

2<\/p>\n

)<\/p>\n

)<\/p>\n

=<\/p>\n

{<\/p>\n

id<\/p>\n

\u2061<\/p>\n

,<\/p>\n

\u03c3<\/p>\n

}<\/p>\n

\\\\boxed{ \\\\operatorname{Aut}(\\\\mathbb Q(\\\\sqrt2)) =\\\\{\\\\operatorname{id},\\\\sigma\\\\} } <\/p>\n

<\/span><\/span><\/span>Aut<\/span><\/span>(<\/span>Q<\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>))<\/span><\/span>=<\/span><\/span>{<\/span>id<\/span><\/span>,<\/span><\/span>\u03c3<\/span>}<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d：<\/p>\n

<\/p>\n

id<\/p>\n

\u2061<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

=<\/p>\n

x<\/p>\n

\\\\operatorname{id}(x)=x <\/p>\n

<\/span><\/span>id<\/span><\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>x<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5e76\u4e14：<\/p>\n

<\/p>\n

\u03c3<\/p>\n

2<\/p>\n

=<\/p>\n

id<\/p>\n

\u2061<\/p>\n

\\\\sigma^2=\\\\operatorname{id} <\/p>\n

<\/span><\/span>\u03c3<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>id<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6240\u4ee5：<\/p>\n

<\/p>\n

L<\/p>\n

\u03c3<\/p>\n

=<\/p>\n

Q<\/p>\n

\\\\boxed{ L^\\\\sigma=\\\\mathbb Q } <\/p>\n

<\/span><\/span><\/span>L<\/span><\/span>\u03c3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u4e3a\u5982\u679c：<\/p>\n

<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

a<\/p>\n

+<\/p>\n

b<\/p>\n

2<\/p>\n

)<\/p>\n

=<\/p>\n

a<\/p>\n

+<\/p>\n

b<\/p>\n

2<\/p>\n

\\\\sigma(a+b\\\\sqrt2)=a+b\\\\sqrt2 <\/p>\n

<\/span><\/span>\u03c3<\/span>(<\/span>a<\/span><\/span>+<\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>a<\/span><\/span>+<\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u90a3\u4e48：<\/p>\n

<\/p>\n

a<\/p>\n

\u2212<\/p>\n

b<\/p>\n

2<\/p>\n

=<\/p>\n

a<\/p>\n

+<\/p>\n

b<\/p>\n

2<\/p>\n

a-b\\\\sqrt2=a+b\\\\sqrt2 <\/p>\n

<\/span><\/span>a<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>a<\/span><\/span>+<\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4ece\u800c：<\/p>\n

<\/p>\n

b<\/p>\n

=<\/p>\n

0<\/p>\n

b=0 <\/p>\n

<\/span><\/span>b<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6240\u4ee5\u4e0d\u52a8\u5143\u7d20\u6070\u597d\u662f\u6709\u7406\u6570\u3002<\/p>\n

\n

<\/p>\n

Q<\/p>\n

(<\/p>\n

2<\/p>\n

,<\/p>\n

3<\/p>\n

)<\/p>\n

\\\\mathbb Q(\\\\sqrt2,\\\\sqrt3)<\/p>\n

<\/span><\/span>Q<\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span> \u7684\u81ea\u540c\u6784<\/h4>\n

<\/p>\n

L<\/p>\n

=<\/p>\n

Q<\/p>\n

(<\/p>\n

2<\/p>\n

,<\/p>\n

3<\/p>\n

)<\/p>\n

L=\\\\mathbb Q(\\\\sqrt2,\\\\sqrt3) <\/p>\n

<\/span><\/span>L<\/span><\/span>=<\/span><\/span><\/span><\/span>Q<\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u4e3a：<\/p>\n

<\/p>\n

6<\/p>\n

=<\/p>\n

2<\/p>\n

3<\/p>\n

\\\\sqrt6=\\\\sqrt2\\\\sqrt3 <\/p>\n

<\/span><\/span><\/span>6<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6240\u4ee5 <\/p>\n

L<\/p>\n

L<\/p>\n

<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span> \u4e2d\u4efb\u610f\u5143\u7d20\u90fd\u53ef\u4ee5\u552f\u4e00\u8868\u793a\u4e3a：<\/p>\n

<\/p>\n

a<\/p>\n

+<\/p>\n

b<\/p>\n

2<\/p>\n

+<\/p>\n

c<\/p>\n

3<\/p>\n

+<\/p>\n

d<\/p>\n

6<\/p>\n

,<\/p>\n

a<\/p>\n

,<\/p>\n

b<\/p>\n

,<\/p>\n

c<\/p>\n

,<\/p>\n

d<\/p>\n

\u2208<\/p>\n

Q<\/p>\n

a+b\\\\sqrt2+c\\\\sqrt3+d\\\\sqrt6, \\\\qquad a,b,c,d\\\\in\\\\mathbb Q <\/p>\n

<\/span><\/span>a<\/span><\/span>+<\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>c<\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>d<\/span><\/span>6<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>a<\/span>,<\/span><\/span>b<\/span>,<\/span><\/span>c<\/span>,<\/span><\/span>d<\/span><\/span>\u2208<\/span><\/span><\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5b9a\u4e49\u81ea\u540c\u6784 <\/p>\n

\u03c3<\/p>\n

\\\\sigma<\/p>\n

<\/span><\/span>\u03c3<\/span><\/span><\/span><\/span><\/span>：<\/p>\n

<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

2<\/p>\n

)<\/p>\n

=<\/p>\n

\u2212<\/p>\n

2<\/p>\n

,<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

3<\/p>\n

)<\/p>\n

=<\/p>\n

3<\/p>\n

\\\\sigma(\\\\sqrt2)=-\\\\sqrt2, \\\\qquad \\\\sigma(\\\\sqrt3)=\\\\sqrt3 <\/p>\n

<\/span><\/span>\u03c3<\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>\u03c3<\/span>(<\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u7531\u4e8e：<\/p>\n

<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

6<\/p>\n

)<\/p>\n

=<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

2<\/p>\n

)<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

3<\/p>\n

)<\/p>\n

=<\/p>\n

\u2212<\/p>\n

6<\/p>\n

\\\\sigma(\\\\sqrt6) =\\\\sigma(\\\\sqrt2)\\\\sigma(\\\\sqrt3) =-\\\\sqrt6 <\/p>\n

<\/span><\/span>\u03c3<\/span>(<\/span><\/span>6<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c3<\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>\u03c3<\/span>(<\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span>6<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u6b64：<\/p>\n

<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

a<\/p>\n

+<\/p>\n

b<\/p>\n

2<\/p>\n

+<\/p>\n

c<\/p>\n

3<\/p>\n

+<\/p>\n

d<\/p>\n

6<\/p>\n

)<\/p>\n

=<\/p>\n

a<\/p>\n

\u2212<\/p>\n

b<\/p>\n

2<\/p>\n

+<\/p>\n

c<\/p>\n

3<\/p>\n

\u2212<\/p>\n

d<\/p>\n

6<\/p>\n

\\\\boxed{ \\\\sigma(a+b\\\\sqrt2+c\\\\sqrt3+d\\\\sqrt6) =a-b\\\\sqrt2+c\\\\sqrt3-d\\\\sqrt6 } <\/p>\n

<\/span><\/span><\/span>\u03c3<\/span>(<\/span>a<\/span><\/span>+<\/span><\/span>b<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>c<\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>d<\/span><\/span>6<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>=<\/span><\/span>a<\/span><\/span>\u2212<\/span><\/span>b<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>c<\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>d<\/span><\/span>6<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u518d\u5b9a\u4e49\u81ea\u540c\u6784 <\/p>\n

\u03c4<\/p>\n

\\\\tau<\/p>\n

<\/span><\/span>\u03c4<\/span><\/span><\/span><\/span><\/span>：<\/p>\n

<\/p>\n

\u03c4<\/p>\n

(<\/p>\n

2<\/p>\n

)<\/p>\n

=<\/p>\n

2<\/p>\n

,<\/p>\n

\u03c4<\/p>\n

(<\/p>\n

3<\/p>\n

)<\/p>\n

=<\/p>\n

\u2212<\/p>\n

3<\/p>\n

\\\\tau(\\\\sqrt2)=\\\\sqrt2, \\\\qquad \\\\tau(\\\\sqrt3)=-\\\\sqrt3 <\/p>\n

<\/span><\/span>\u03c4<\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>\u03c4<\/span>(<\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4e8e\u662f：<\/p>\n

<\/p>\n

\u03c4<\/p>\n

(<\/p>\n

6<\/p>\n

)<\/p>\n

=<\/p>\n

\u03c4<\/p>\n

(<\/p>\n

2<\/p>\n

)<\/p>\n

\u03c4<\/p>\n

(<\/p>\n

3<\/p>\n

)<\/p>\n

=<\/p>\n

\u2212<\/p>\n

6<\/p>\n

\\\\tau(\\\\sqrt6) =\\\\tau(\\\\sqrt2)\\\\tau(\\\\sqrt3) =-\\\\sqrt6 <\/p>\n

<\/span><\/span>\u03c4<\/span>(<\/span><\/span>6<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c4<\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>\u03c4<\/span>(<\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span>6<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u6b64：<\/p>\n

<\/p>\n

\u03c4<\/p>\n

(<\/p>\n

a<\/p>\n

+<\/p>\n

b<\/p>\n

2<\/p>\n

+<\/p>\n

c<\/p>\n

3<\/p>\n

+<\/p>\n

d<\/p>\n

6<\/p>\n

)<\/p>\n

=<\/p>\n

a<\/p>\n

+<\/p>\n

b<\/p>\n

2<\/p>\n

\u2212<\/p>\n

c<\/p>\n

3<\/p>\n

\u2212<\/p>\n

d<\/p>\n

6<\/p>\n

\\\\boxed{ \\\\tau(a+b\\\\sqrt2+c\\\\sqrt3+d\\\\sqrt6) =a+b\\\\sqrt2-c\\\\sqrt3-d\\\\sqrt6 } <\/p>\n

<\/span><\/span><\/span>\u03c4<\/span>(<\/span>a<\/span><\/span>+<\/span><\/span>b<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>c<\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>d<\/span><\/span>6<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>=<\/span><\/span>a<\/span><\/span>+<\/span><\/span>b<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>c<\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>d<\/span><\/span>6<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u8fd8\u53ef\u4ee5\u540c\u65f6\u6539\u53d8 <\/p>\n

2<\/p>\n

\\\\sqrt2<\/p>\n

<\/span><\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u548c <\/p>\n

3<\/p>\n

\\\\sqrt3<\/p>\n

<\/span><\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u7684\u7b26\u53f7：<\/p>\n

<\/p>\n

\u03c3<\/p>\n

\u03c4<\/p>\n

(<\/p>\n

2<\/p>\n

)<\/p>\n

=<\/p>\n

\u2212<\/p>\n

2<\/p>\n

\\\\sigma\\\\tau(\\\\sqrt2)=-\\\\sqrt2 <\/p>\n

<\/span><\/span>\u03c3<\/span>\u03c4<\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4ee5\u53ca：<\/p>\n

<\/p>\n

\u03c3<\/p>\n

\u03c4<\/p>\n

(<\/p>\n

3<\/p>\n

)<\/p>\n

=<\/p>\n

\u2212<\/p>\n

3<\/p>\n

\\\\sigma\\\\tau(\\\\sqrt3)=-\\\\sqrt3 <\/p>\n

<\/span><\/span>\u03c3<\/span>\u03c4<\/span>(<\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6240\u4ee5：<\/p>\n

<\/p>\n

\u03c3<\/p>\n

\u03c4<\/p>\n

(<\/p>\n

6<\/p>\n

)<\/p>\n

=<\/p>\n

6<\/p>\n

\\\\sigma\\\\tau(\\\\sqrt6)=\\\\sqrt6 <\/p>\n

<\/span><\/span>\u03c3<\/span>\u03c4<\/span>(<\/span><\/span>6<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>6<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4ece\u800c：<\/p>\n

<\/p>\n

\u03c3<\/p>\n

\u03c4<\/p>\n

(<\/p>\n

a<\/p>\n

+<\/p>\n

b<\/p>\n

2<\/p>\n

+<\/p>\n

c<\/p>\n

3<\/p>\n

+<\/p>\n

d<\/p>\n

6<\/p>\n

)<\/p>\n

=<\/p>\n

a<\/p>\n

\u2212<\/p>\n

b<\/p>\n

2<\/p>\n

\u2212<\/p>\n

c<\/p>\n

3<\/p>\n

+<\/p>\n

d<\/p>\n

6<\/p>\n

\\\\boxed{\\\\sigma\\\\tau(a+b\\\\sqrt2+c\\\\sqrt3+d\\\\sqrt6) =a-b\\\\sqrt2-c\\\\sqrt3+d\\\\sqrt6 } <\/p>\n

<\/span><\/span><\/span>\u03c3<\/span>\u03c4<\/span>(<\/span>a<\/span><\/span>+<\/span><\/span>b<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>c<\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>d<\/span><\/span>6<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>=<\/span><\/span>a<\/span><\/span>\u2212<\/span><\/span>b<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>c<\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>d<\/span><\/span>6<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u6b64\u4e00\u5171\u6709\u56db\u4e2a\u81ea\u540c\u6784：<\/p>\n

<\/p>\n

Aut<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

)<\/p>\n

=<\/p>\n

{<\/p>\n

id<\/p>\n

\u2061<\/p>\n

,<\/p>\n

\u03c3<\/p>\n

,<\/p>\n

\u03c4<\/p>\n

,<\/p>\n

\u03c3<\/p>\n

\u03c4<\/p>\n

}<\/p>\n

\\\\boxed{ \\\\operatorname{Aut}(L) =\\\\{ \\\\operatorname{id}, \\\\sigma, \\\\tau, \\\\sigma\\\\tau \\\\} } <\/p>\n

<\/span><\/span><\/span>Aut<\/span><\/span>(<\/span>L<\/span>)<\/span><\/span>=<\/span><\/span>{<\/span>id<\/span><\/span>,<\/span><\/span>\u03c3<\/span>,<\/span><\/span>\u03c4<\/span>,<\/span><\/span>\u03c3<\/span>\u03c4<\/span>}<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\n
1. \u8fd9\u4e2a\u81ea\u540c\u6784\u7fa4\u662f\u514b\u83b1\u56e0\u56db\u5143\u7fa4<\/h5>\n

\u8fd9\u56db\u4e2a\u81ea\u540c\u6784\u6ee1\u8db3：<\/p>\n

<\/p>\n

\u03c3<\/p>\n

2<\/p>\n

=<\/p>\n

\u03c4<\/p>\n

2<\/p>\n

=<\/p>\n

id<\/p>\n

\u2061<\/p>\n

\\\\sigma^2=\\\\tau^2=\\\\operatorname{id} <\/p>\n

<\/span><\/span>\u03c3<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c4<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>id<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5e76\u4e14：<\/p>\n

<\/p>\n

\u03c3<\/p>\n

\u03c4<\/p>\n

=<\/p>\n

\u03c4<\/p>\n

\u03c3<\/p>\n

\\\\sigma\\\\tau=\\\\tau\\\\sigma <\/p>\n

<\/span><\/span>\u03c3<\/span>\u03c4<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c4<\/span>\u03c3<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u540c\u65f6：<\/p>\n

<\/p>\n

(<\/p>\n

\u03c3<\/p>\n

\u03c4<\/p>\n

)<\/p>\n

2<\/p>\n

=<\/p>\n

id<\/p>\n

\u2061<\/p>\n

(\\\\sigma\\\\tau)^2=\\\\operatorname{id} <\/p>\n

<\/span><\/span>(<\/span>\u03c3<\/span>\u03c4<\/span>)<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>id<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u6b64\u4e09\u4e2a\u975e\u5355\u4f4d\u5143：<\/p>\n

<\/p>\n

\u03c3<\/p>\n

,<\/p>\n

\u03c4<\/p>\n

,<\/p>\n

\u03c3<\/p>\n

\u03c4<\/p>\n

\\\\sigma,\\\\quad\\\\tau,\\\\quad\\\\sigma\\\\tau <\/p>\n

<\/span><\/span>\u03c3<\/span>,<\/span><\/span><\/span>\u03c4<\/span>,<\/span><\/span><\/span>\u03c3<\/span>\u03c4<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u7684\u9636\u5168\u90e8\u90fd\u662f <\/p>\n

2<\/p>\n

2<\/p>\n

<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

\u6240\u4ee5\u8fd9\u4e2a\u7fa4\u4e0d\u662f\u5faa\u73af\u7fa4 <\/p>\n

C<\/p>\n

4<\/p>\n

C_4<\/p>\n

<\/span><\/span>C<\/span><\/span>4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u800c\u662f：<\/p>\n

<\/p>\n

Aut<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

)<\/p>\n

\u2245<\/p>\n

C<\/p>\n

2<\/p>\n

\u00d7<\/p>\n

C<\/p>\n

2<\/p>\n

\\\\boxed{ \\\\operatorname{Aut}(L)\\\\cong C_2\\\\times C_2 } <\/p>\n

<\/span><\/span><\/span>Aut<\/span><\/span>(<\/span>L<\/span>)<\/span><\/span>\u2245<\/span><\/span>C<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00d7<\/span><\/span>C<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u8fd9\u4e2a\u7fa4\u5c31\u662f\u514b\u83b1\u56e0\u56db\u5143\u7fa4，\u901a\u5e38\u8bb0\u4f5c：<\/p>\n

<\/p>\n

V<\/p>\n

4<\/p>\n

\\\\boxed{V_4} <\/p>\n

<\/span><\/span><\/span>V<\/span><\/span>4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u6b64：<\/p>\n

<\/p>\n

Aut<\/p>\n

\u2061<\/p>\n

(<\/p>\n

Q<\/p>\n

(<\/p>\n

2<\/p>\n

,<\/p>\n

3<\/p>\n

)<\/p>\n

)<\/p>\n

\u2245<\/p>\n

V<\/p>\n

4<\/p>\n

\\\\boxed{ \\\\operatorname{Aut}(\\\\mathbb Q(\\\\sqrt2,\\\\sqrt3)) \\\\cong V_4 } <\/p>\n

<\/span><\/span><\/span>Aut<\/span><\/span>(<\/span>Q<\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>))<\/span><\/span>\u2245<\/span><\/span>V<\/span><\/span>4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5b83\u7684\u7fa4\u7ed3\u6784\u53ef\u4ee5\u5199\u6210：<\/p>\n

<\/p>\n

\u81ea\u540c\u6784<\/p>\n

(<\/p>\n

2<\/p>\n

,<\/p>\n

3<\/p>\n

)<\/p>\n

\u00a0\u7684\u53d8\u5316<\/p>\n

id<\/p>\n

\u2061<\/p>\n

(<\/p>\n

2<\/p>\n

,<\/p>\n

3<\/p>\n

)<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

\u2212<\/p>\n

2<\/p>\n

,<\/p>\n

3<\/p>\n

)<\/p>\n

\u03c4<\/p>\n

(<\/p>\n

2<\/p>\n

,<\/p>\n

\u2212<\/p>\n

3<\/p>\n

)<\/p>\n

\u03c3<\/p>\n

\u03c4<\/p>\n

(<\/p>\n

\u2212<\/p>\n

2<\/p>\n

,<\/p>\n

\u2212<\/p>\n

3<\/p>\n

)<\/p>\n

\\\\begin{array}{c|c} \\\\text{\u81ea\u540c\u6784} & (\\\\sqrt2,\\\\sqrt3)\\\\text{ \u7684\u53d8\u5316}\\\\\\\\ \\\\hline \\\\operatorname{id} & (\\\\sqrt2,\\\\sqrt3) \\\\\\\\[2mm] \\\\sigma & (-\\\\sqrt2,\\\\sqrt3) \\\\\\\\[2mm] \\\\tau & (\\\\sqrt2,-\\\\sqrt3) \\\\\\\\[2mm] \\\\sigma\\\\tau & (-\\\\sqrt2,-\\\\sqrt3) \\\\end{array} <\/p>\n

<\/span><\/span><\/span><\/span><\/span>\u81ea\u540c\u6784<\/span><\/span><\/span><\/span><\/span>id<\/span><\/span><\/span><\/span><\/span>\u03c3<\/span><\/span><\/span><\/span>\u03c4<\/span><\/span><\/span><\/span>\u03c3<\/span>\u03c4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>\u00a0<\/span>\u7684\u53d8\u5316<\/span><\/span><\/span><\/span><\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>(<\/span>\u2212<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u2212<\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>(<\/span>\u2212<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u2212<\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u53ef\u4ee5\u628a\u5b83\u7406\u89e3\u6210：<\/p>\n

2<\/p>\n

\\\\sqrt2<\/p>\n

<\/span><\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u548c <\/p>\n

3<\/p>\n

\\\\sqrt3<\/p>\n

<\/span><\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u7684\u7b26\u53f7\u5206\u522b\u6709\u4e24\u79cd\u9009\u62e9，\u56e0\u6b64\u4e00\u5171\u6709：<\/p>\n

<\/p>\n

2<\/p>\n

\u00d7<\/p>\n

2<\/p>\n

=<\/p>\n

4<\/p>\n

2\\\\times2=4 <\/p>\n

<\/span><\/span>2<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>2<\/span><\/span>=<\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u79cd\u53ef\u80fd\u3002<\/p>\n

\n
2. <\/p>\n

\u03c3<\/p>\n

\\\\sigma<\/p>\n

<\/span><\/span>\u03c3<\/span><\/span><\/span><\/span><\/span> \u7684\u4e0d\u52a8\u57df<\/h5>\n

\u8003\u8651：<\/p>\n

<\/p>\n

L<\/p>\n

\u03c3<\/p>\n

=<\/p>\n

{<\/p>\n

x<\/p>\n

\u2208<\/p>\n

L<\/p>\n

:<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

=<\/p>\n

x<\/p>\n

}<\/p>\n

L^\\\\sigma =\\\\{x\\\\in L:\\\\sigma(x)=x\\\\} <\/p>\n

<\/span><\/span>L<\/span><\/span>\u03c3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>{<\/span>x<\/span><\/span>\u2208<\/span><\/span><\/span><\/span>L<\/span><\/span>:<\/span><\/span><\/span><\/span>\u03c3<\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>x<\/span>}<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u8bbe：<\/p>\n

<\/p>\n

x<\/p>\n

=<\/p>\n

a<\/p>\n

+<\/p>\n

b<\/p>\n

2<\/p>\n

+<\/p>\n

c<\/p>\n

3<\/p>\n

+<\/p>\n

d<\/p>\n

6<\/p>\n

x=a+b\\\\sqrt2+c\\\\sqrt3+d\\\\sqrt6 <\/p>\n

<\/span><\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span>a<\/span><\/span>+<\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>c<\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>d<\/span><\/span>6<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5982\u679c <\/p>\n

x<\/p>\n

x<\/p>\n

<\/span><\/span>x<\/span><\/span><\/span><\/span><\/span> \u662f <\/p>\n

\u03c3<\/p>\n

\\\\sigma<\/p>\n

<\/span><\/span>\u03c3<\/span><\/span><\/span><\/span><\/span> \u7684\u4e0d\u52a8\u5143\u7d20，\u5219：<\/p>\n

<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

=<\/p>\n

x<\/p>\n

\\\\sigma(x)=x <\/p>\n

<\/span><\/span>\u03c3<\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>x<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5373：<\/p>\n

<\/p>\n

a<\/p>\n

\u2212<\/p>\n

b<\/p>\n

2<\/p>\n

+<\/p>\n

c<\/p>\n

3<\/p>\n

\u2212<\/p>\n

d<\/p>\n

6<\/p>\n

=<\/p>\n

a<\/p>\n

+<\/p>\n

b<\/p>\n

2<\/p>\n

+<\/p>\n

c<\/p>\n

3<\/p>\n

+<\/p>\n

d<\/p>\n

6<\/p>\n

a-b\\\\sqrt2+c\\\\sqrt3-d\\\\sqrt6 =a+b\\\\sqrt2+c\\\\sqrt3+d\\\\sqrt6 <\/p>\n

<\/span><\/span>a<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>c<\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>d<\/span><\/span>6<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>a<\/span><\/span>+<\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>c<\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>d<\/span><\/span>6<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6240\u4ee5：<\/p>\n

<\/p>\n

b<\/p>\n

=<\/p>\n

d<\/p>\n

=<\/p>\n

0<\/p>\n

b=d=0 <\/p>\n

<\/span><\/span>b<\/span><\/span>=<\/span><\/span><\/span><\/span>d<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u6b64：<\/p>\n

<\/p>\n

x<\/p>\n

=<\/p>\n

a<\/p>\n

+<\/p>\n

c<\/p>\n

3<\/p>\n

x=a+c\\\\sqrt3 <\/p>\n

<\/span><\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span>a<\/span><\/span>+<\/span><\/span><\/span><\/span>c<\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4ece\u800c：<\/p>\n

<\/p>\n

L<\/p>\n

\u03c3<\/p>\n

=<\/p>\n

Q<\/p>\n

(<\/p>\n

3<\/p>\n

)<\/p>\n

\\\\boxed{ L^\\\\sigma=\\\\mathbb Q(\\\\sqrt3) } <\/p>\n

<\/span><\/span><\/span>L<\/span><\/span>\u03c3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>Q<\/span>(<\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\n
3. <\/p>\n

\u03c4<\/p>\n

\\\\tau<\/p>\n

<\/span><\/span>\u03c4<\/span><\/span><\/span><\/span><\/span> \u7684\u4e0d\u52a8\u57df<\/h5>\n

\u540c\u7406：<\/p>\n

<\/p>\n

L<\/p>\n

\u03c4<\/p>\n

=<\/p>\n

{<\/p>\n

x<\/p>\n

\u2208<\/p>\n

L<\/p>\n

:<\/p>\n

\u03c4<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

=<\/p>\n

x<\/p>\n

}<\/p>\n

L^\\\\tau =\\\\{x\\\\in L:\\\\tau(x)=x\\\\} <\/p>\n

<\/span><\/span>L<\/span><\/span>\u03c4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>{<\/span>x<\/span><\/span>\u2208<\/span><\/span><\/span><\/span>L<\/span><\/span>:<\/span><\/span><\/span><\/span>\u03c4<\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>x<\/span>}<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u8bbe：<\/p>\n

<\/p>\n

x<\/p>\n

=<\/p>\n

a<\/p>\n

+<\/p>\n

b<\/p>\n

2<\/p>\n

+<\/p>\n

c<\/p>\n

3<\/p>\n

+<\/p>\n

d<\/p>\n

6<\/p>\n

x=a+b\\\\sqrt2+c\\\\sqrt3+d\\\\sqrt6 <\/p>\n

<\/span><\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span>a<\/span><\/span>+<\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>c<\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>d<\/span><\/span>6<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u7531：<\/p>\n

<\/p>\n

\u03c4<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

=<\/p>\n

x<\/p>\n

\\\\tau(x)=x <\/p>\n

<\/span><\/span>\u03c4<\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>x<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5f97\u5230：<\/p>\n

<\/p>\n

c<\/p>\n

=<\/p>\n

d<\/p>\n

=<\/p>\n

0<\/p>\n

c=d=0 <\/p>\n

<\/span><\/span>c<\/span><\/span>=<\/span><\/span><\/span><\/span>d<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u6b64：<\/p>\n

<\/p>\n

x<\/p>\n

=<\/p>\n

a<\/p>\n

+<\/p>\n

b<\/p>\n

2<\/p>\n

x=a+b\\\\sqrt2 <\/p>\n

<\/span><\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span>a<\/span><\/span>+<\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6240\u4ee5：<\/p>\n

<\/p>\n

L<\/p>\n

\u03c4<\/p>\n

=<\/p>\n

Q<\/p>\n

(<\/p>\n

2<\/p>\n

)<\/p>\n

\\\\boxed{ L^\\\\tau=\\\\mathbb Q(\\\\sqrt2) } <\/p>\n

<\/span><\/span><\/span>L<\/span><\/span>\u03c4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>Q<\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\n
4. <\/p>\n

\u03c3<\/p>\n

\u03c4<\/p>\n

\\\\sigma\\\\tau<\/p>\n

<\/span><\/span>\u03c3<\/span>\u03c4<\/span><\/span><\/span><\/span><\/span> \u7684\u4e0d\u52a8\u57df<\/h5>\n

\u73b0\u5728\u8003\u8651：<\/p>\n

<\/p>\n

L<\/p>\n

\u03c3<\/p>\n

\u03c4<\/p>\n

L^{\\\\sigma\\\\tau} <\/p>\n

<\/span><\/span>L<\/span><\/span>\u03c3<\/span>\u03c4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u4e3a：<\/p>\n

<\/p>\n

\u03c3<\/p>\n

\u03c4<\/p>\n

(<\/p>\n

2<\/p>\n

)<\/p>\n

=<\/p>\n

\u2212<\/p>\n

2<\/p>\n

\\\\sigma\\\\tau(\\\\sqrt2)=-\\\\sqrt2 <\/p>\n

<\/span><\/span>\u03c3<\/span>\u03c4<\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4ee5\u53ca：<\/p>\n

<\/p>\n

\u03c3<\/p>\n

\u03c4<\/p>\n

(<\/p>\n

3<\/p>\n

)<\/p>\n

=<\/p>\n

\u2212<\/p>\n

3<\/p>\n

\\\\sigma\\\\tau(\\\\sqrt3)=-\\\\sqrt3 <\/p>\n

<\/span><\/span>\u03c3<\/span>\u03c4<\/span>(<\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4f46\u662f：<\/p>\n

<\/p>\n

\u03c3<\/p>\n

\u03c4<\/p>\n

(<\/p>\n

6<\/p>\n

)<\/p>\n

=<\/p>\n

6<\/p>\n

\\\\sigma\\\\tau(\\\\sqrt6)=\\\\sqrt6 <\/p>\n

<\/span><\/span>\u03c3<\/span>\u03c4<\/span>(<\/span><\/span>6<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>6<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6240\u4ee5：<\/p>\n

<\/p>\n

\u03c3<\/p>\n

\u03c4<\/p>\n

(<\/p>\n

a<\/p>\n

+<\/p>\n

b<\/p>\n

2<\/p>\n

+<\/p>\n

c<\/p>\n

3<\/p>\n

+<\/p>\n

d<\/p>\n

6<\/p>\n

)<\/p>\n

=<\/p>\n

a<\/p>\n

\u2212<\/p>\n

b<\/p>\n

2<\/p>\n

\u2212<\/p>\n

c<\/p>\n

3<\/p>\n

+<\/p>\n

d<\/p>\n

6<\/p>\n

\\\\sigma\\\\tau (a+b\\\\sqrt2+c\\\\sqrt3+d\\\\sqrt6) =a-b\\\\sqrt2-c\\\\sqrt3+d\\\\sqrt6 <\/p>\n

<\/span><\/span>\u03c3<\/span>\u03c4<\/span>(<\/span>a<\/span><\/span>+<\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>c<\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>d<\/span><\/span>6<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>a<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>c<\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>d<\/span><\/span>6<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u82e5：<\/p>\n

<\/p>\n

\u03c3<\/p>\n

\u03c4<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

=<\/p>\n

x<\/p>\n

\\\\sigma\\\\tau(x)=x <\/p>\n

<\/span><\/span>\u03c3<\/span>\u03c4<\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>x<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5219：<\/p>\n

<\/p>\n

b<\/p>\n

=<\/p>\n

c<\/p>\n

=<\/p>\n

0<\/p>\n

b=c=0 <\/p>\n

<\/span><\/span>b<\/span><\/span>=<\/span><\/span><\/span><\/span>c<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u6b64：<\/p>\n

<\/p>\n

x<\/p>\n

=<\/p>\n

a<\/p>\n

+<\/p>\n

d<\/p>\n

6<\/p>\n

x=a+d\\\\sqrt6 <\/p>\n

<\/span><\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span>a<\/span><\/span>+<\/span><\/span><\/span><\/span>d<\/span><\/span>6<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4ece\u800c：<\/p>\n

<\/p>\n

L<\/p>\n

\u03c3<\/p>\n

\u03c4<\/p>\n

=<\/p>\n

Q<\/p>\n

(<\/p>\n

6<\/p>\n

)<\/p>\n

\\\\boxed{ L^{\\\\sigma\\\\tau}=\\\\mathbb Q(\\\\sqrt6) } <\/p>\n

<\/span><\/span><\/span>L<\/span><\/span>\u03c3<\/span>\u03c4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>Q<\/span>(<\/span><\/span>6<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\n
5. \u4e09\u4e2a\u975e\u5e73\u51e1\u81ea\u540c\u6784\u4e0e\u4e09\u4e2a\u4e2d\u95f4\u57df<\/h5>\n

\u56e0\u6b64：<\/p>\n

<\/p>\n

\u5b50\u7fa4<\/p>\n

\u4e0d\u52a8\u57df<\/p>\n

\u27e8<\/p>\n

\u03c3<\/p>\n

\u27e9<\/p>\n

Q<\/p>\n

(<\/p>\n

3<\/p>\n

)<\/p>\n

\u27e8<\/p>\n

\u03c4<\/p>\n

\u27e9<\/p>\n

Q<\/p>\n

(<\/p>\n

2<\/p>\n

)<\/p>\n

\u27e8<\/p>\n

\u03c3<\/p>\n

\u03c4<\/p>\n

\u27e9<\/p>\n

Q<\/p>\n

(<\/p>\n

6<\/p>\n

)<\/p>\n

\\\\boxed{ \\\\begin{array}{c|c} \\\\text{\u5b50\u7fa4} & \\\\text{\u4e0d\u52a8\u57df}\\\\\\\\ \\\\hline \\\\langle\\\\sigma\\\\rangle & \\\\mathbb Q(\\\\sqrt3) \\\\\\\\[2mm] \\\\langle\\\\tau\\\\rangle & \\\\mathbb Q(\\\\sqrt2) \\\\\\\\[2mm] \\\\langle\\\\sigma\\\\tau\\\\rangle & \\\\mathbb Q(\\\\sqrt6) \\\\end{array} } <\/p>\n

<\/span><\/span><\/span><\/span><\/span><\/span>\u5b50\u7fa4<\/span><\/span><\/span><\/span><\/span>\u27e8<\/span>\u03c3<\/span>\u27e9<\/span><\/span><\/span><\/span>\u27e8<\/span>\u03c4<\/span>\u27e9<\/span><\/span><\/span><\/span>\u27e8<\/span>\u03c3<\/span>\u03c4<\/span>\u27e9<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4e0d\u52a8\u57df<\/span><\/span><\/span><\/span><\/span>Q<\/span>(<\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>Q<\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>Q<\/span>(<\/span><\/span>6<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u800c\u6574\u4e2a\u7fa4\u7684\u4e0d\u52a8\u57df\u662f：<\/p>\n

<\/p>\n

L<\/p>\n

Aut<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

)<\/p>\n

=<\/p>\n

Q<\/p>\n

L^{\\\\operatorname{Aut}(L)} =\\\\mathbb Q <\/p>\n

<\/span><\/span>L<\/span><\/span>Aut<\/span><\/span>(<\/span>L<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u8fd9\u8bf4\u660e：<\/p>\n

<\/p>\n

Q<\/p>\n

\u2282<\/p>\n

Q<\/p>\n

(<\/p>\n

2<\/p>\n

)<\/p>\n

,<\/p>\n

Q<\/p>\n

(<\/p>\n

3<\/p>\n

)<\/p>\n

,<\/p>\n

Q<\/p>\n

(<\/p>\n

6<\/p>\n

)<\/p>\n

\u2282<\/p>\n

Q<\/p>\n

(<\/p>\n

2<\/p>\n

,<\/p>\n

3<\/p>\n

)<\/p>\n

\\\\mathbb Q \\\\subset \\\\mathbb Q(\\\\sqrt2),\\\\mathbb Q(\\\\sqrt3),\\\\mathbb Q(\\\\sqrt6) \\\\subset \\\\mathbb Q(\\\\sqrt2,\\\\sqrt3) <\/p>\n

<\/span><\/span>Q<\/span><\/span>\u2282<\/span><\/span><\/span><\/span>Q<\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>,<\/span><\/span>Q<\/span>(<\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>,<\/span><\/span>Q<\/span>(<\/span><\/span>6<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>\u2282<\/span><\/span><\/span><\/span>Q<\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6784\u6210\u4e86\u4e00\u4e2a\u5178\u578b\u7684\u4e2d\u95f4\u57df\u7ed3\u6784\u3002<\/p>\n

\n
6. \u4e0e\u4f3d\u7f57\u74e6\u7fa4\u7684\u5173\u7cfb<\/h5>\n

\u7531\u4e8e：<\/p>\n

<\/p>\n

[<\/p>\n

Q<\/p>\n

(<\/p>\n

2<\/p>\n

,<\/p>\n

3<\/p>\n

)<\/p>\n

:<\/p>\n

Q<\/p>\n

]<\/p>\n

=<\/p>\n

4<\/p>\n

[\\\\mathbb Q(\\\\sqrt2,\\\\sqrt3):\\\\mathbb Q]=4 <\/p>\n

<\/span><\/span>[<\/span>Q<\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>:<\/span><\/span><\/span><\/span>Q<\/span>]<\/span><\/span>=<\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u800c\u81ea\u540c\u6784\u7684\u6570\u91cf\u4e5f\u662f：<\/p>\n

<\/p>\n

\u2223<\/p>\n

Aut<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

)<\/p>\n

\u2223<\/p>\n

=<\/p>\n

4<\/p>\n

|\\\\operatorname{Aut}(L)|=4 <\/p>\n

<\/span><\/span>\u2223<\/span><\/span>Aut<\/span><\/span>(<\/span>L<\/span>)<\/span>\u2223<\/span><\/span>=<\/span><\/span><\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u6b64：<\/p>\n

<\/p>\n

\u2223<\/p>\n

Aut<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

)<\/p>\n

\u2223<\/p>\n

=<\/p>\n

[<\/p>\n

L<\/p>\n

:<\/p>\n

Q<\/p>\n

]<\/p>\n

|\\\\operatorname{Aut}(L)| =[L:\\\\mathbb Q] <\/p>\n

<\/span><\/span>\u2223<\/span><\/span>Aut<\/span><\/span>(<\/span>L<\/span>)<\/span>\u2223<\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span>L<\/span><\/span>:<\/span><\/span><\/span><\/span>Q<\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6240\u4ee5：<\/p>\n

<\/p>\n

Q<\/p>\n

(<\/p>\n

2<\/p>\n

,<\/p>\n

3<\/p>\n

)<\/p>\n

\/<\/p>\n

Q<\/p>\n

\u00a0\u662f\u4f3d\u7f57\u74e6\u6269\u5f20<\/p>\n

\\\\boxed{ \\\\mathbb Q(\\\\sqrt2,\\\\sqrt3)\/\\\\mathbb Q \\\\text{ \u662f\u4f3d\u7f57\u74e6\u6269\u5f20} } <\/p>\n

<\/span><\/span><\/span>Q<\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>\/<\/span>Q<\/span>\u00a0<\/span>\u662f\u4f3d\u7f57\u74e6\u6269\u5f20<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4f3d\u7f57\u74e6\u7fa4\u4e3a：<\/p>\n

<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

Q<\/p>\n

(<\/p>\n

2<\/p>\n

,<\/p>\n

3<\/p>\n

)<\/p>\n

\/<\/p>\n

Q<\/p>\n

)<\/p>\n

=<\/p>\n

Aut<\/p>\n

\u2061<\/p>\n

(<\/p>\n

Q<\/p>\n

(<\/p>\n

2<\/p>\n

,<\/p>\n

3<\/p>\n

)<\/p>\n

)<\/p>\n

\u2245<\/p>\n

V<\/p>\n

4<\/p>\n

\\\\boxed{ \\\\operatorname{Gal} (\\\\mathbb Q(\\\\sqrt2,\\\\sqrt3)\/\\\\mathbb Q) =\\\\operatorname{Aut} (\\\\mathbb Q(\\\\sqrt2,\\\\sqrt3)) \\\\cong V_4 } <\/p>\n

<\/span><\/span><\/span>Gal<\/span><\/span>(<\/span>Q<\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>\/<\/span>Q<\/span>)<\/span><\/span>=<\/span><\/span>Aut<\/span><\/span>(<\/span>Q<\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>))<\/span><\/span>\u2245<\/span><\/span>V<\/span><\/span>4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u8fd9\u4e2a\u4f8b\u5b50\u5b8c\u6574\u5730\u5c55\u793a\u4e86：<\/p>\n

<\/p>\n

\u57df\u6269\u5f20<\/p>\n

\u27f6<\/p>\n

\u81ea\u540c\u6784<\/p>\n

\u27f6<\/p>\n

\u4e0d\u52a8\u57df<\/p>\n

\u27f6<\/p>\n

\u4f3d\u7f57\u74e6\u7fa4<\/p>\n

\\\\boxed{ \\\\text{\u57df\u6269\u5f20} \\\\longrightarrow \\\\text{\u81ea\u540c\u6784} \\\\longrightarrow \\\\text{\u4e0d\u52a8\u57df} \\\\longrightarrow \\\\text{\u4f3d\u7f57\u74e6\u7fa4} } <\/p>\n

<\/span><\/span><\/span>\u57df\u6269\u5f20<\/span><\/span><\/span>\u27f6<\/span><\/span>\u81ea\u540c\u6784<\/span><\/span><\/span>\u27f6<\/span><\/span>\u4e0d\u52a8\u57df<\/span><\/span><\/span>\u27f6<\/span><\/span>\u4f3d\u7f57\u74e6\u7fa4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5e76\u4e14\u5728\u8fd9\u4e2a\u4f8b\u5b50\u4e2d，\u514b\u83b1\u56e0\u56db\u5143\u7fa4 <\/p>\n

V<\/p>\n

4<\/p>\n

V_4<\/p>\n

<\/span><\/span>V<\/span><\/span>4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u7684\u4e09\u4e2a\u975e\u5e73\u51e1\u5b50\u7fa4，\u6070\u597d\u5bf9\u5e94\u4e09\u4e2a\u4e2d\u95f4\u57df：<\/p>\n

<\/p>\n

Q<\/p>\n

(<\/p>\n

2<\/p>\n

,<\/p>\n

3<\/p>\n

)<\/p>\n

\/<\/p>\n

\u2223<\/p>\n

\\\\<\/p>\n

Q<\/p>\n

(<\/p>\n

2<\/p>\n

)<\/p>\n

Q<\/p>\n

(<\/p>\n

3<\/p>\n

)<\/p>\n

Q<\/p>\n

(<\/p>\n

6<\/p>\n

)<\/p>\n

\\\\<\/p>\n

\u2223<\/p>\n

\/<\/p>\n

Q<\/p>\n

\\\\boxed{ \\\\begin{array}{ccc} &\\\\mathbb Q(\\\\sqrt2,\\\\sqrt3)&\\\\\\\\ \/&\\\\vert&\\\\backslash\\\\\\\\ \\\\mathbb Q(\\\\sqrt2)& \\\\mathbb Q(\\\\sqrt3)& \\\\mathbb Q(\\\\sqrt6)\\\\\\\\ \\\\backslash&\\\\vert&\/\\\\\\\\ &\\\\mathbb Q& \\\\end{array} } <\/p>\n

<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\/<\/span><\/span><\/span><\/span>Q<\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>\\\\<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Q<\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>\u2223<\/span><\/span><\/span><\/span>Q<\/span>(<\/span><\/span>3<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>\u2223<\/span><\/span><\/span><\/span>Q<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\\\\<\/span><\/span><\/span><\/span>Q<\/span>(<\/span><\/span>6<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>\/<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\n

\u4f3d\u7f57\u74e6\u81ea\u540c\u6784<\/h2>\n

\u57df\u81ea\u540c\u6784\u662f <\/p>\n

L<\/p>\n

L<\/p>\n

<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span> \u5230\u81ea\u8eab\u7684\u57df\u81ea\u540c\u6784：<\/p>\n

<\/p>\n

\u03c3<\/p>\n

\u2208<\/p>\n

Aut<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

)<\/p>\n

\\\\sigma\\\\in\\\\operatorname{Aut}(L) <\/p>\n

<\/span><\/span>\u03c3<\/span><\/span>\u2208<\/span><\/span><\/span><\/span>Aut<\/span><\/span>(<\/span>L<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5982\u679c\u8fdb\u4e00\u6b65\u8981\u6c42 <\/p>\n

\u03c3<\/p>\n

\\\\sigma<\/p>\n

<\/span><\/span>\u03c3<\/span><\/span><\/span><\/span><\/span> \u56fa\u5b9a\u57fa\u57df <\/p>\n

K<\/p>\n

K<\/p>\n

<\/span><\/span>K<\/span><\/span><\/span><\/span><\/span> \u4e2d\u7684\u6240\u6709\u5143\u7d20：<\/p>\n

<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

a<\/p>\n

)<\/p>\n

=<\/p>\n

a<\/p>\n

,<\/p>\n

\u2200<\/p>\n

a<\/p>\n

\u2208<\/p>\n

K<\/p>\n

\\\\sigma(a)=a, \\\\qquad \\\\forall a\\\\in K <\/p>\n

<\/span><\/span>\u03c3<\/span>(<\/span>a<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>a<\/span>,<\/span><\/span><\/span>\u2200<\/span>a<\/span><\/span>\u2208<\/span><\/span><\/span><\/span>K<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u90a3\u4e48 <\/p>\n

\u03c3<\/p>\n

\\\\sigma<\/p>\n

<\/span><\/span>\u03c3<\/span><\/span><\/span><\/span><\/span> \u5c31\u662f <\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

L\/K<\/p>\n

<\/span><\/span>L<\/span>\/<\/span>K<\/span><\/span><\/span><\/span><\/span> \u7684\u4f3d\u7f57\u74e6\u81ea\u540c\u6784\u3002<\/p>\n

\u56e0\u6b64：<\/p>\n

<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

)<\/p>\n

=<\/p>\n

{<\/p>\n

\u03c3<\/p>\n

\u2208<\/p>\n

Aut<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

)<\/p>\n

\u2223<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

a<\/p>\n

)<\/p>\n

=<\/p>\n

a<\/p>\n

,<\/p>\n

\u00a0<\/p>\n

\u2200<\/p>\n

a<\/p>\n

\u2208<\/p>\n

K<\/p>\n

}<\/p>\n

\\\\boxed{\\\\operatorname{Gal}(L\/K) =\\\\{\\\\sigma\\\\in\\\\operatorname{Aut}(L) \\\\mid \\\\sigma(a)=a,\\\\ \\\\forall a\\\\in K \\\\} } <\/p>\n

<\/span><\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>K<\/span>)<\/span><\/span>=<\/span><\/span>{<\/span>\u03c3<\/span><\/span>\u2208<\/span><\/span>Aut<\/span><\/span>(<\/span>L<\/span>)<\/span><\/span>\u2223<\/span><\/span>\u03c3<\/span>(<\/span>a<\/span>)<\/span><\/span>=<\/span><\/span>a<\/span>,<\/span>\u00a0<\/span><\/span>\u2200<\/span>a<\/span><\/span>\u2208<\/span><\/span>K<\/span>}<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6240\u4ee5：<\/p>\n

<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

)<\/p>\n

\u2264<\/p>\n

Aut<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

)<\/p>\n

\\\\boxed{ \\\\operatorname{Gal}(L\/K) \\\\leq \\\\operatorname{Aut}(L) } <\/p>\n

<\/span><\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>K<\/span>)<\/span><\/span>\u2264<\/span><\/span>Aut<\/span><\/span>(<\/span>L<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5373：<\/p>\n

<\/p>\n

\u57df\u81ea\u540c\u6784<\/p>\n

\u2287<\/p>\n

\u56fa\u5b9a\u00a0\u57fa\u57df<\/p>\n

K<\/p>\n

\u00a0\u7684\u81ea\u540c\u6784<\/p>\n

=<\/p>\n

\u4f3d\u7f57\u74e6\u81ea\u540c\u6784<\/p>\n

\\\\boxed{ \\\\text{\u57df\u81ea\u540c\u6784} \\\\supseteq\\\\text{\u56fa\u5b9a }\u57fa\u57dfK\\\\text{ \u7684\u81ea\u540c\u6784} =\\\\text{\u4f3d\u7f57\u74e6\u81ea\u540c\u6784} } <\/p>\n

<\/span><\/span><\/span>\u57df\u81ea\u540c\u6784<\/span><\/span><\/span>\u2287<\/span><\/span>\u56fa\u5b9a<\/span>\u00a0<\/span><\/span>\u57fa\u57df<\/span>K<\/span>\u00a0<\/span>\u7684\u81ea\u540c\u6784<\/span><\/span><\/span>=<\/span><\/span>\u4f3d\u7f57\u74e6\u81ea\u540c\u6784<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5355\u4e2a\u81ea\u540c\u6784\u7684\u4e0d\u52a8\u57df<\/h2>\n

\u8bbe <\/p>\n

L<\/p>\n

L<\/p>\n

<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span> \u662f\u4e00\u4e2a\u57df，<\/p>\n

\u03c3<\/p>\n

\u2208<\/p>\n

Aut<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

)<\/p>\n

\\\\sigma\\\\in\\\\operatorname{Aut}(L)<\/p>\n

<\/span><\/span>\u03c3<\/span><\/span>\u2208<\/span><\/span><\/span><\/span>Aut<\/span><\/span>(<\/span>L<\/span>)<\/span><\/span><\/span><\/span><\/span> \u662f <\/p>\n

L<\/p>\n

L<\/p>\n

<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span> \u7684\u4e00\u4e2a\u81ea\u540c\u6784\u3002<\/p>\n

\u5b9a\u4e49 <\/p>\n

\u03c3<\/p>\n

\\\\sigma<\/p>\n

<\/span><\/span>\u03c3<\/span><\/span><\/span><\/span><\/span> \u7684\u4e0d\u52a8\u57df\u4e3a：<\/p>\n

<\/p>\n

L<\/p>\n

\u03c3<\/p>\n

=<\/p>\n

{<\/p>\n

x<\/p>\n

\u2208<\/p>\n

L<\/p>\n

\u2223<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

=<\/p>\n

x<\/p>\n

}<\/p>\n

\\\\boxed{ L^\\\\sigma= \\\\{x\\\\in L\\\\mid \\\\sigma(x)=x\\\\} } <\/p>\n

<\/span><\/span><\/span>L<\/span><\/span>\u03c3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>{<\/span>x<\/span><\/span>\u2208<\/span><\/span>L<\/span><\/span>\u2223<\/span><\/span>\u03c3<\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span>x<\/span>}<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> <\/p>\n

L<\/p>\n

\u03c3<\/p>\n

\u2286<\/p>\n

L<\/p>\n

L^\\\\sigma\\\\subseteq L<\/p>\n

<\/span><\/span>L<\/span><\/span>\u03c3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2286<\/span><\/span><\/span><\/span>L<\/span><\/span><\/span><\/span><\/span><\/p>\n

<\/p>\n

L<\/p>\n

\u03c3<\/p>\n

L^\\\\sigma<\/p>\n

<\/span><\/span>L<\/span><\/span>\u03c3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u662f\u4ece <\/p>\n

L<\/p>\n

L<\/p>\n

<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span> \u4e2d\u6311\u51fa\u6765\u7684\u3001\u88ab <\/p>\n

\u03c3<\/p>\n

\\\\sigma<\/p>\n

<\/span><\/span>\u03c3<\/span><\/span><\/span><\/span><\/span> \u56fa\u5b9a\u7684\u5143\u7d20：<\/p>\n

<\/p>\n

L<\/p>\n

\u03c3<\/p>\n

L^\\\\sigma<\/p>\n

<\/span><\/span>L<\/span><\/span>\u03c3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u662f <\/p>\n

L<\/p>\n

L<\/p>\n

<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span> \u7684\u5b50\u57df, <\/p>\n

L<\/p>\n

L<\/p>\n

<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span>\u662f <\/p>\n

L<\/p>\n

\u03c3<\/p>\n

L^\\\\sigma<\/p>\n

<\/span><\/span>L<\/span><\/span>\u03c3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u6269\u57df<\/p>\n

\u4f8b\u5982：<\/p>\n

<\/p>\n

L<\/p>\n

=<\/p>\n

Q<\/p>\n

(<\/p>\n

2<\/p>\n

)<\/p>\n

L=\\\\mathbb{Q}(\\\\sqrt2)<\/p>\n

<\/span><\/span>L<\/span><\/span>=<\/span><\/span><\/span><\/span>Q<\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5b9a\u4e49：<\/p>\n

<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

2<\/p>\n

)<\/p>\n

=<\/p>\n

\u2212<\/p>\n

2<\/p>\n

\\\\sigma(\\\\sqrt2)=-\\\\sqrt2<\/p>\n

<\/span><\/span>\u03c3<\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u90a3\u4e48：<\/p>\n

<\/p>\n

L<\/p>\n

\u03c3<\/p>\n

=<\/p>\n

Q<\/p>\n

L^\\\\sigma=\\\\mathbb{Q}<\/p>\n

<\/span><\/span>L<\/span><\/span>\u03c3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span><\/p>\n

<\/p>\n

Q<\/p>\n

=<\/p>\n

L<\/p>\n

\u03c3<\/p>\n

\u2282<\/p>\n

L<\/p>\n

=<\/p>\n

Q<\/p>\n

(<\/p>\n

2<\/p>\n

)<\/p>\n

\\\\mathbb{Q}=L^\\\\sigma\\\\subset L=\\\\mathbb{Q}(\\\\sqrt2)<\/p>\n

<\/span><\/span>Q<\/span><\/span>=<\/span><\/span><\/span><\/span>L<\/span><\/span>\u03c3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2282<\/span><\/span><\/span><\/span>L<\/span><\/span>=<\/span><\/span><\/span><\/span>Q<\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/p>\n

\u81ea\u540c\u6784\u5b50\u7fa4\u7684\u4e0d\u52a8\u57df<\/h2>\n

\u8bbe <\/p>\n

G<\/p>\n

\u2264<\/p>\n

Aut<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

)<\/p>\n

G\\\\leq\\\\operatorname{Aut}(L)<\/p>\n

<\/span><\/span>G<\/span><\/span>\u2264<\/span><\/span><\/span><\/span>Aut<\/span><\/span>(<\/span>L<\/span>)<\/span><\/span><\/span><\/span><\/span> \u662f <\/p>\n

L<\/p>\n

L<\/p>\n

<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span> \u7684\u4e00\u4e2a\u81ea\u540c\u6784\u5b50\u7fa4\u3002<\/p>\n

\u5b9a\u4e49 <\/p>\n

G<\/p>\n

G<\/p>\n

<\/span><\/span>G<\/span><\/span><\/span><\/span><\/span> \u7684\u4e0d\u52a8\u57df\u4e3a：<\/p>\n

<\/p>\n

L<\/p>\n

G<\/p>\n

=<\/p>\n

{<\/p>\n

x<\/p>\n

\u2208<\/p>\n

L<\/p>\n

\u2223<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

=<\/p>\n

x<\/p>\n

,<\/p>\n

\u00a0<\/p>\n

\u2200<\/p>\n

\u03c3<\/p>\n

\u2208<\/p>\n

G<\/p>\n

}<\/p>\n

\\\\boxed{ L^G= \\\\{x\\\\in L\\\\mid \\\\sigma(x)=x,\\\\ \\\\forall\\\\sigma\\\\in G \\\\} } <\/p>\n

<\/span><\/span><\/span>L<\/span><\/span>G<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>{<\/span>x<\/span><\/span>\u2208<\/span><\/span>L<\/span><\/span>\u2223<\/span><\/span>\u03c3<\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span>x<\/span>,<\/span>\u00a0<\/span><\/span>\u2200<\/span>\u03c3<\/span><\/span>\u2208<\/span><\/span>G<\/span>}<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

<\/p>\n

L<\/p>\n

G<\/p>\n

=<\/p>\n

\u22c2<\/p>\n

\u03c3<\/p>\n

\u2208<\/p>\n

G<\/p>\n

L<\/p>\n

\u03c3<\/p>\n

\\\\boxed{ L^G=\\\\bigcap_{\\\\sigma\\\\in G}L^\\\\sigma } <\/p>\n

<\/span><\/span><\/span>L<\/span><\/span>G<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span>\u03c3<\/span>\u2208<\/span>G<\/span><\/span><\/span><\/span><\/span>\u22c2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>L<\/span><\/span>\u03c3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4f3d\u7f57\u74e6\u81ea\u540c\u6784\u4e0e\u4e0d\u52a8\u57df\u7684\u5173\u7cfb<\/h2>\n

\u5bf9\u4e8e：<\/p>\n

<\/p>\n

\u03c3<\/p>\n

\u2208<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

)<\/p>\n

\\\\sigma\\\\in\\\\operatorname{Gal}(L\/K) <\/p>\n

<\/span><\/span>\u03c3<\/span><\/span>\u2208<\/span><\/span><\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>K<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u4e3a <\/p>\n

\u03c3<\/p>\n

\\\\sigma<\/p>\n

<\/span><\/span>\u03c3<\/span><\/span><\/span><\/span><\/span> \u56fa\u5b9a <\/p>\n

K<\/p>\n

K<\/p>\n

<\/span><\/span>K<\/span><\/span><\/span><\/span><\/span> \u4e2d\u7684\u6240\u6709\u5143\u7d20，\u6240\u4ee5：<\/p>\n

<\/p>\n

K<\/p>\n

\u2286<\/p>\n

L<\/p>\n

\u03c3<\/p>\n

K\\\\subseteq L^\\\\sigma <\/p>\n

<\/span><\/span>K<\/span><\/span>\u2286<\/span><\/span><\/span><\/span>L<\/span><\/span>\u03c3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d：<\/p>\n

<\/p>\n

L<\/p>\n

\u03c3<\/p>\n

=<\/p>\n

{<\/p>\n

x<\/p>\n

\u2208<\/p>\n

L<\/p>\n

\u2223<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

=<\/p>\n

x<\/p>\n

}<\/p>\n

L^\\\\sigma= \\\\{x\\\\in L\\\\mid\\\\sigma(x)=x\\\\} <\/p>\n

<\/span><\/span>L<\/span><\/span>\u03c3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>{<\/span>x<\/span><\/span>\u2208<\/span><\/span><\/span><\/span>L<\/span><\/span>\u2223<\/span><\/span><\/span><\/span>\u03c3<\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>x<\/span>}<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5982\u679c\u4e00\u4e2a\u81ea\u540c\u6784\u5b50\u7fa4：<\/p>\n

<\/p>\n

G<\/p>\n

\u2264<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

)<\/p>\n

G\\\\leq\\\\operatorname{Gal}(L\/K) <\/p>\n

<\/span><\/span>G<\/span><\/span>\u2264<\/span><\/span><\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>K<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u90a3\u4e48 <\/p>\n

G<\/p>\n

G<\/p>\n

<\/span><\/span>G<\/span><\/span><\/span><\/span><\/span> \u7684\u4e0d\u52a8\u57df\u4e3a：<\/p>\n

<\/p>\n

L<\/p>\n

G<\/p>\n

=<\/p>\n

{<\/p>\n

x<\/p>\n

\u2208<\/p>\n

L<\/p>\n

\u2223<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

=<\/p>\n

x<\/p>\n

,<\/p>\n

\u00a0<\/p>\n

\u2200<\/p>\n

\u03c3<\/p>\n

\u2208<\/p>\n

G<\/p>\n

}<\/p>\n

\\\\boxed{ L^G= \\\\{x\\\\in L\\\\mid \\\\sigma(x)=x,\\\\ \\\\forall\\\\sigma\\\\in G \\\\} } <\/p>\n

<\/span><\/span><\/span>L<\/span><\/span>G<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>{<\/span>x<\/span><\/span>\u2208<\/span><\/span>L<\/span><\/span>\u2223<\/span><\/span>\u03c3<\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span>x<\/span>,<\/span>\u00a0<\/span><\/span>\u2200<\/span>\u03c3<\/span><\/span>\u2208<\/span><\/span>G<\/span>}<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5e76\u4e14：<\/p>\n

<\/p>\n

K<\/p>\n

\u2286<\/p>\n

L<\/p>\n

G<\/p>\n

\u2286<\/p>\n

L<\/p>\n

\\\\boxed{ K\\\\subseteq L^G\\\\subseteq L } <\/p>\n

<\/span><\/span><\/span>K<\/span><\/span>\u2286<\/span><\/span>L<\/span><\/span>G<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2286<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4f3d\u7f57\u74e6\u5bf9\u5e94<\/h2>\n

\u8bbe <\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

L\/K<\/p>\n

<\/span><\/span>L<\/span>\/<\/span>K<\/span><\/span><\/span><\/span><\/span> \u662f\u4e00\u4e2a\u4f3d\u7f57\u74e6\u6269\u5f20\u3002<\/p>\n

\u4f3d\u7f57\u74e6\u7fa4\u4e3a：<\/p>\n

<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

)<\/p>\n

\\\\operatorname{Gal}(L\/K) <\/p>\n

<\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>K<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4f3d\u7f57\u74e6\u5bf9\u5e94\u5efa\u7acb\u4e86：<\/p>\n

<\/p>\n

\u4e2d\u95f4\u57df<\/p>\n

\u27f7<\/p>\n

\u4f3d\u7f57\u74e6\u7fa4\u7684\u5b50\u7fa4<\/p>\n

\\\\boxed{ \\\\text{\u4e2d\u95f4\u57df} \\\\quad\\\\longleftrightarrow\\\\quad \\\\text{\u4f3d\u7f57\u74e6\u7fa4\u7684\u5b50\u7fa4} } <\/p>\n

<\/span><\/span><\/span>\u4e2d\u95f4\u57df<\/span><\/span><\/span><\/span>\u27f7<\/span><\/span><\/span>\u4f3d\u7f57\u74e6\u7fa4\u7684\u5b50\u7fa4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d\u4e2d\u95f4\u57df\u662f\u6ee1\u8db3：<\/p>\n

<\/p>\n

K<\/p>\n

\u2286<\/p>\n

F<\/p>\n

\u2286<\/p>\n

L<\/p>\n

K\\\\subseteq F\\\\subseteq L <\/p>\n

<\/span><\/span>K<\/span><\/span>\u2286<\/span><\/span><\/span><\/span>F<\/span><\/span>\u2286<\/span><\/span><\/span><\/span>L<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u7684\u57df <\/p>\n

F<\/p>\n

F<\/p>\n

<\/span><\/span>F<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

\n

\u4ece\u5b50\u7fa4\u5f97\u5230\u4e0d\u52a8\u57df<\/h4>\n

\u5bf9\u4e8e\u4efb\u610f\u5b50\u7fa4：<\/p>\n

<\/p>\n

G<\/p>\n

\u2264<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

)<\/p>\n

G\\\\leq\\\\operatorname{Gal}(L\/K) <\/p>\n

<\/span><\/span>G<\/span><\/span>\u2264<\/span><\/span><\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>K<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u53ef\u4ee5\u5f97\u5230\u4e00\u4e2a\u4e0d\u52a8\u57df：<\/p>\n

<\/p>\n

L<\/p>\n

G<\/p>\n

=<\/p>\n

{<\/p>\n

x<\/p>\n

\u2208<\/p>\n

L<\/p>\n

\u2223<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

=<\/p>\n

x<\/p>\n

,<\/p>\n

\u00a0<\/p>\n

\u2200<\/p>\n

\u03c3<\/p>\n

\u2208<\/p>\n

G<\/p>\n

}<\/p>\n

\\\\boxed{ L^G= \\\\{x\\\\in L\\\\mid \\\\sigma(x)=x,\\\\ \\\\forall\\\\sigma\\\\in G \\\\} } <\/p>\n

<\/span><\/span><\/span>L<\/span><\/span>G<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>{<\/span>x<\/span><\/span>\u2208<\/span><\/span>L<\/span><\/span>\u2223<\/span><\/span>\u03c3<\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span>x<\/span>,<\/span>\u00a0<\/span><\/span>\u2200<\/span>\u03c3<\/span><\/span>\u2208<\/span><\/span>G<\/span>}<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5e76\u4e14：<\/p>\n

<\/p>\n

K<\/p>\n

\u2286<\/p>\n

L<\/p>\n

G<\/p>\n

\u2286<\/p>\n

L<\/p>\n

\\\\boxed{ K\\\\subseteq L^G\\\\subseteq L } <\/p>\n

<\/span><\/span><\/span>K<\/span><\/span>\u2286<\/span><\/span>L<\/span><\/span>G<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2286<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6240\u4ee5：<\/p>\n

<\/p>\n

G<\/p>\n

\u27f6<\/p>\n

L<\/p>\n

G<\/p>\n

G \\\\quad\\\\longrightarrow\\\\quad L^G <\/p>\n

<\/span><\/span>G<\/span><\/span><\/span>\u27f6<\/span><\/span><\/span><\/span><\/span>L<\/span><\/span>G<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\n

\u4ece\u4e2d\u95f4\u57df\u5f97\u5230\u5b50\u7fa4<\/h4>\n

\u5bf9\u4e8e\u4efb\u610f\u4e2d\u95f4\u57df：<\/p>\n

<\/p>\n

K<\/p>\n

\u2286<\/p>\n

F<\/p>\n

\u2286<\/p>\n

L<\/p>\n

K\\\\subseteq F\\\\subseteq L <\/p>\n

<\/span><\/span>K<\/span><\/span>\u2286<\/span><\/span><\/span><\/span>F<\/span><\/span>\u2286<\/span><\/span><\/span><\/span>L<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5b9a\u4e49：<\/p>\n

<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

F<\/p>\n

)<\/p>\n

=<\/p>\n

{<\/p>\n

\u03c3<\/p>\n

\u2208<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

)<\/p>\n

\u2223<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

=<\/p>\n

x<\/p>\n

,<\/p>\n

\u00a0<\/p>\n

\u2200<\/p>\n

x<\/p>\n

\u2208<\/p>\n

F<\/p>\n

}<\/p>\n

\\\\boxed{ \\\\operatorname{Gal}(L\/F) =\\\\{\\\\sigma\\\\in\\\\operatorname{Gal}(L\/K) \\\\mid \\\\sigma(x)=x,\\\\ \\\\forall x\\\\in F \\\\} } <\/p>\n

<\/span><\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>F<\/span>)<\/span><\/span>=<\/span><\/span>{<\/span>\u03c3<\/span><\/span>\u2208<\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>K<\/span>)<\/span><\/span>\u2223<\/span><\/span>\u03c3<\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span>x<\/span>,<\/span>\u00a0<\/span><\/span>\u2200<\/span>x<\/span><\/span>\u2208<\/span><\/span>F<\/span>}<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4e5f\u5c31\u662f\u8bf4：<\/p>\n

<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

F<\/p>\n

)<\/p>\n

\\\\operatorname{Gal}(L\/F)<\/p>\n

<\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>F<\/span>)<\/span><\/span><\/span><\/span><\/span> \u662f\u6240\u6709\u56fa\u5b9a <\/p>\n

F<\/p>\n

F<\/p>\n

<\/span><\/span>F<\/span><\/span><\/span><\/span><\/span> \u4e2d\u5143\u7d20\u7684\u4f3d\u7f57\u74e6\u81ea\u540c\u6784\u7ec4\u6210\u7684\u5b50\u7fa4\u3002<\/p>\n

\u6240\u4ee5：<\/p>\n

<\/p>\n

F<\/p>\n

\u27f6<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

F<\/p>\n

)<\/p>\n

F \\\\quad\\\\longrightarrow\\\\quad \\\\operatorname{Gal}(L\/F) <\/p>\n

<\/span><\/span>F<\/span><\/span><\/span>\u27f6<\/span><\/span><\/span><\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>F<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\n

\u4e24\u4e2a\u5bf9\u5e94\u5173\u7cfb<\/h4>\n

\u56e0\u6b64：<\/p>\n

<\/p>\n

G<\/p>\n

\u27f7<\/p>\n

L<\/p>\n

G<\/p>\n

\\\\boxed{ G \\\\quad\\\\longleftrightarrow\\\\quad L^G } <\/p>\n

<\/span><\/span><\/span>G<\/span><\/span><\/span>\u27f7<\/span><\/span><\/span>L<\/span><\/span>G<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4ee5\u53ca：<\/p>\n

<\/p>\n

F<\/p>\n

\u27f7<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

F<\/p>\n

)<\/p>\n

\\\\boxed{ F \\\\quad\\\\longleftrightarrow\\\\quad \\\\operatorname{Gal}(L\/F) } <\/p>\n

<\/span><\/span><\/span>F<\/span><\/span><\/span>\u27f7<\/span><\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>F<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u66f4\u5b8c\u6574\u5730\u5199：<\/p>\n

<\/p>\n

{<\/p>\n

K<\/p>\n

\u2286<\/p>\n

F<\/p>\n

\u2286<\/p>\n

L<\/p>\n

}<\/p>\n

\u27f7<\/p>\n

{<\/p>\n

G<\/p>\n

\u2264<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

)<\/p>\n

}<\/p>\n

\\\\boxed{ \\\\left\\\\{ \\\\begin{array}{c} K\\\\subseteq F\\\\subseteq L \\\\end{array} \\\\right\\\\} \\\\quad \\\\longleftrightarrow \\\\quad \\\\left\\\\{ \\\\begin{array}{c} G\\\\leq\\\\operatorname{Gal}(L\/K) \\\\end{array} \\\\right\\\\} } <\/p>\n

<\/span><\/span><\/span>{<\/span><\/span><\/span><\/span>K<\/span><\/span>\u2286<\/span><\/span>F<\/span><\/span>\u2286<\/span><\/span>L<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>}<\/span><\/span><\/span><\/span><\/span>\u27f7<\/span><\/span><\/span>{<\/span><\/span><\/span><\/span>G<\/span><\/span>\u2264<\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>K<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>}<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d\u5bf9\u5e94\u5173\u7cfb\u4e3a：<\/p>\n

<\/p>\n

F<\/p>\n

=<\/p>\n

L<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

F<\/p>\n

)<\/p>\n

\\\\boxed{ F=L^{\\\\operatorname{Gal}(L\/F)} } <\/p>\n

<\/span><\/span><\/span>F<\/span><\/span>=<\/span><\/span>L<\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>F<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4ee5\u53ca：<\/p>\n

<\/p>\n

G<\/p>\n

=<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

L<\/p>\n

G<\/p>\n

)<\/p>\n

\\\\boxed{ G=\\\\operatorname{Gal}(L\/L^G) } <\/p>\n

<\/span><\/span><\/span>G<\/span><\/span>=<\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>L<\/span><\/span>G<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u6b64，\u4f3d\u7f57\u74e6\u5bf9\u5e94\u7684\u6838\u5fc3\u5c31\u662f：<\/p>\n

<\/p>\n

\u4e2d\u95f4\u57df<\/p>\n

\u27f7<\/p>\n

\u5b50\u7fa4<\/p>\n

\\\\boxed{ \\\\text{\u4e2d\u95f4\u57df} \\\\quad\\\\longleftrightarrow\\\\quad \\\\text{\u5b50\u7fa4} } <\/p>\n

<\/span><\/span><\/span>\u4e2d\u95f4\u57df<\/span><\/span><\/span><\/span>\u27f7<\/span><\/span><\/span>\u5b50\u7fa4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u901a\u8fc7\u201c\u4e0d\u52a8\u57df\u201d\u5efa\u7acb\u4e8c\u8005\u4e4b\u95f4\u7684\u5bf9\u5e94\u3002<\/p>\n

\u53ef\u5206\u6269\u5f20<\/h2>\n

\u8bbe <\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

L\/K<\/p>\n

<\/span><\/span>L<\/span>\/<\/span>K<\/span><\/span><\/span><\/span><\/span> \u662f\u4e00\u4e2a\u4ee3\u6570\u6269\u5f20\u3002<\/p>\n

\u5982\u679c <\/p>\n

L<\/p>\n

L<\/p>\n

<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span> \u4e2d\u7684\u6bcf\u4e00\u4e2a\u5143\u7d20 <\/p>\n

\u03b1<\/p>\n

\\\\alpha<\/p>\n

<\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span> \u5728 <\/p>\n

K<\/p>\n

K<\/p>\n

<\/span><\/span>K<\/span><\/span><\/span><\/span><\/span> \u4e0a\u7684\u6700\u5c0f\u591a\u9879\u5f0f\u90fd\u6ca1\u6709\u91cd\u590d\u6839，\u90a3\u4e48\u79f0 <\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

L\/K<\/p>\n

<\/span><\/span>L<\/span>\/<\/span>K<\/span><\/span><\/span><\/span><\/span> \u662f\u53ef\u5206\u6269\u5f20\u3002<\/p>\n

\u5373：<\/p>\n

<\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

\u00a0\u662f\u53ef\u5206\u6269\u5f20
\n \u2005\u200a<\/p>\n

\u27fa
\n \u2005\u200a<\/p>\n

\u2200<\/p>\n

\u03b1<\/p>\n

\u2208<\/p>\n

L<\/p>\n

,<\/p>\n

\u00a0<\/p>\n

m<\/p>\n

\u03b1<\/p>\n

,<\/p>\n

K<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

\u00a0\u6ca1\u6709\u91cd\u590d\u6839<\/p>\n

\\\\boxed{ L\/K\\\\text{ \u662f\u53ef\u5206\u6269\u5f20} \\\\iff \\\\forall\\\\alpha\\\\in L,\\\\ m_{\\\\alpha,K}(x)\\\\text{ \u6ca1\u6709\u91cd\u590d\u6839} } <\/p>\n

<\/span><\/span><\/span>L<\/span>\/<\/span>K<\/span>\u00a0<\/span>\u662f\u53ef\u5206\u6269\u5f20<\/span><\/span><\/span><\/span>\u27fa<\/span><\/span><\/span>\u2200<\/span>\u03b1<\/span><\/span>\u2208<\/span><\/span>L<\/span>,<\/span>\u00a0<\/span><\/span>m<\/span><\/span>\u03b1<\/span>,<\/span>K<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span>)<\/span>\u00a0<\/span>\u6ca1\u6709\u91cd\u590d\u6839<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d <\/p>\n

m<\/p>\n

\u03b1<\/p>\n

,<\/p>\n

K<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

m_{\\\\alpha,K}(x)<\/p>\n

<\/span><\/span>m<\/span><\/span>\u03b1<\/span>,<\/span>K<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span> \u8868\u793a <\/p>\n

\u03b1<\/p>\n

\\\\alpha<\/p>\n

<\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span> \u5728 <\/p>\n

K<\/p>\n

K<\/p>\n

<\/span><\/span>K<\/span><\/span><\/span><\/span><\/span> \u4e0a\u7684\u6700\u5c0f\u591a\u9879\u5f0f\u3002<\/p>\n

\u53ef\u5206\u6269\u5f20\u7684\u4f8b\u5b50<\/h3>\n

<\/p>\n

L<\/p>\n

=<\/p>\n

Q<\/p>\n

(<\/p>\n

2<\/p>\n

)<\/p>\n

,<\/p>\n

K<\/p>\n

=<\/p>\n

Q<\/p>\n

L=\\\\mathbb{Q}(\\\\sqrt2),\\\\qquad K=\\\\mathbb{Q} <\/p>\n

<\/span><\/span>L<\/span><\/span>=<\/span><\/span><\/span><\/span>Q<\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>,<\/span><\/span><\/span>K<\/span><\/span>=<\/span><\/span><\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

<\/p>\n

2<\/p>\n

\\\\sqrt2<\/p>\n

<\/span><\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u5728 <\/p>\n

Q<\/p>\n

\\\\mathbb{Q}<\/p>\n

<\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span> \u4e0a\u7684\u6700\u5c0f\u591a\u9879\u5f0f\u4e3a：<\/p>\n

<\/p>\n

x<\/p>\n

2<\/p>\n

\u2212<\/p>\n

2<\/p>\n

x^2-2 <\/p>\n

<\/span><\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5b83\u7684\u4e24\u4e2a\u6839\u4e3a：<\/p>\n

<\/p>\n

2<\/p>\n

,<\/p>\n

\u2212<\/p>\n

2<\/p>\n

\\\\sqrt2,\\\\quad-\\\\sqrt2 <\/p>\n

<\/span><\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>\u2212<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4e24\u4e2a\u6839\u4e0d\u540c，\u56e0\u6b64\u6ca1\u6709\u91cd\u590d\u6839\u3002<\/p>\n

\u4e0d\u53ef\u5206\u6269\u5f20\u7684\u4f8b\u5b50<\/h3>\n

\u4e0d\u53ef\u5206\u6269\u5f20\u901a\u5e38\u51fa\u73b0\u5728\u6b63\u7279\u5f81\u7684\u57df\u4e2d\u3002<\/p>\n

\u8bbe：<\/p>\n

<\/p>\n

K<\/p>\n

=<\/p>\n

F<\/p>\n

p<\/p>\n

(<\/p>\n

t<\/p>\n

)<\/p>\n

K=\\\\mathbb{F}_p(t)<\/p>\n

<\/span><\/span>K<\/span><\/span>=<\/span><\/span><\/span><\/span>F<\/span><\/span>p<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d <\/p>\n

p<\/p>\n

p<\/p>\n

<\/span><\/span>p<\/span><\/span><\/span><\/span><\/span> \u662f\u7d20\u6570，<\/p>\n

t<\/p>\n

t<\/p>\n

<\/span><\/span>t<\/span><\/span><\/span><\/span><\/span> \u662f\u4e00\u4e2a\u4e0d\u5b9a\u5143\u3002<\/p>\n

\u4ee4 <\/p>\n

\u03b1<\/p>\n

\\\\alpha<\/p>\n

<\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span> \u6ee1\u8db3：<\/p>\n

<\/p>\n

\u03b1<\/p>\n

p<\/p>\n

=<\/p>\n

t<\/p>\n

\\\\alpha^p=t <\/p>\n

<\/span><\/span>\u03b1<\/span><\/span>p<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5e76\u4ee4：<\/p>\n

<\/p>\n

L<\/p>\n

=<\/p>\n

K<\/p>\n

(<\/p>\n

\u03b1<\/p>\n

)<\/p>\n

L=K(\\\\alpha) <\/p>\n

<\/span><\/span>L<\/span><\/span>=<\/span><\/span><\/span><\/span>K<\/span>(<\/span>\u03b1<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u90a3\u4e48 <\/p>\n

\u03b1<\/p>\n

\\\\alpha<\/p>\n

<\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span> \u5728 <\/p>\n

K<\/p>\n

K<\/p>\n

<\/span><\/span>K<\/span><\/span><\/span><\/span><\/span> \u4e0a\u7684\u6700\u5c0f\u591a\u9879\u5f0f\u4e3a：<\/p>\n

<\/p>\n

f<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

=<\/p>\n

x<\/p>\n

p<\/p>\n

\u2212<\/p>\n

t<\/p>\n

f(x)=x^p-t <\/p>\n

<\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>x<\/span><\/span>p<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4f46\u662f：<\/p>\n

<\/p>\n

f<\/p>\n

\u2032<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

=<\/p>\n

p<\/p>\n

x<\/p>\n

p<\/p>\n

\u2212<\/p>\n

1<\/p>\n

=<\/p>\n

0<\/p>\n

f'(x)=px^{p-1}=0 <\/p>\n

<\/span><\/span>f<\/span><\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>p<\/span>x<\/span><\/span>p<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u6b64 <\/p>\n

f<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

f(x)<\/p>\n

<\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span> \u6709\u91cd\u6839\u3002<\/p>\n

\u4e8b\u5b9e\u4e0a，\u5728\u4ee3\u6570\u95ed\u5305\u4e2d：<\/p>\n

<\/p>\n

x<\/p>\n

p<\/p>\n

\u2212<\/p>\n

t<\/p>\n

=<\/p>\n

(<\/p>\n

x<\/p>\n

\u2212<\/p>\n

\u03b1<\/p>\n

)<\/p>\n

p<\/p>\n

x^p-t=(x-\\\\alpha)^p <\/p>\n

<\/span><\/span>x<\/span><\/span>p<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>t<\/span><\/span>=<\/span><\/span><\/span><\/span>(<\/span>x<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03b1<\/span>)<\/span><\/span>p<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6240\u4ee5 <\/p>\n

\u03b1<\/p>\n

\\\\alpha<\/p>\n

<\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span> \u662f <\/p>\n

p<\/p>\n

p<\/p>\n

<\/span><\/span>p<\/span><\/span><\/span><\/span><\/span> \u91cd\u6839\u3002<\/p>\n

\u56e0\u6b64：<\/p>\n

<\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

\u00a0\u4e0d\u662f\u53ef\u5206\u6269\u5f20<\/p>\n

\\\\boxed{ L\/K\\\\text{ \u4e0d\u662f\u53ef\u5206\u6269\u5f20} } <\/p>\n

<\/span><\/span><\/span>L<\/span>\/<\/span>K<\/span>\u00a0<\/span>\u4e0d\u662f\u53ef\u5206\u6269\u5f20<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u6216 <\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

\u00a0\u662f\u4e0d\u53ef\u5206\u6269\u5f20<\/p>\n

\\\\boxed{ L\/K\\\\text{ \u662f\u4e0d\u53ef\u5206\u6269\u5f20} } <\/p>\n

<\/span><\/span><\/span>L<\/span>\/<\/span>K<\/span>\u00a0<\/span>\u662f\u4e0d\u53ef\u5206\u6269\u5f20<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u8fd9\u4e2a\u4f8b\u5b50\u8bf4\u660e：<\/p>\n

<\/p>\n

\u4e0d\u53ef\u5206\u6269\u5f20\u53ef\u4ee5\u7531\u6700\u5c0f\u591a\u9879\u5f0f\u5177\u6709\u91cd\u6839\u4ea7\u751f<\/p>\n

\\\\boxed{ \\\\text{\u4e0d\u53ef\u5206\u6269\u5f20\u53ef\u4ee5\u7531\u6700\u5c0f\u591a\u9879\u5f0f\u5177\u6709\u91cd\u6839\u4ea7\u751f} } <\/p>\n

<\/span><\/span><\/span>\u4e0d\u53ef\u5206\u6269\u5f20\u53ef\u4ee5\u7531\u6700\u5c0f\u591a\u9879\u5f0f\u5177\u6709\u91cd\u6839\u4ea7\u751f<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6b63\u89c4\u6269\u5f20<\/h2>\n

\u8bbe <\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

L\/K<\/p>\n

<\/span><\/span>L<\/span>\/<\/span>K<\/span><\/span><\/span><\/span><\/span> \u662f\u4e00\u4e2a\u4ee3\u6570\u6269\u5f20\u3002<\/p>\n

\u5982\u679c <\/p>\n

K<\/p>\n

[<\/p>\n

x<\/p>\n

]<\/p>\n

K[x]<\/p>\n

<\/span><\/span>K<\/span>[<\/span>x<\/span>]<\/span><\/span><\/span><\/span><\/span> \u4e2d\u4efb\u610f\u4e00\u4e2a\u5728 <\/p>\n

L<\/p>\n

L<\/p>\n

<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span> \u4e2d\u6709\u6839\u7684\u4e0d\u53ef\u7ea6\u591a\u9879\u5f0f，\u53ea\u8981\u5b83\u5728 <\/p>\n

L<\/p>\n

L<\/p>\n

<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span> \u4e2d\u6709\u4e00\u4e2a\u6839，\u90a3\u4e48\u5b83\u7684\u6240\u6709\u6839\u90fd\u5728 <\/p>\n

L<\/p>\n

L<\/p>\n

<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span> \u4e2d，\u5219\u79f0 <\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

L\/K<\/p>\n

<\/span><\/span>L<\/span>\/<\/span>K<\/span><\/span><\/span><\/span><\/span> \u662f\u6b63\u89c4\u6269\u5f20\u3002<\/p>\n

\u76f4\u89c2\u7406\u89e3：<\/p>\n

\u4e00\u4e2a\u4e0d\u53ef\u7ea6\u591a\u9879\u5f0f\u7684\u6839\u53ea\u8981\u6709\u4e00\u4e2a\u843d\u5728 <\/p>\n

L<\/p>\n

L<\/p>\n

<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span> \u91cc\u9762，\u90a3\u4e48\u5b83\u7684\u5176\u4ed6\u6839\u4e5f\u5fc5\u987b\u5168\u90e8\u843d\u5728 <\/p>\n

L<\/p>\n

L<\/p>\n

<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span> \u91cc\u9762\u3002<\/p>\n

\u6b63\u89c4\u6269\u5f20\u7684\u4f8b\u5b50<\/h3>\n

<\/p>\n

L<\/p>\n

=<\/p>\n

Q<\/p>\n

(<\/p>\n

2<\/p>\n

)<\/p>\n

,<\/p>\n

K<\/p>\n

=<\/p>\n

Q<\/p>\n

L=\\\\mathbb{Q}(\\\\sqrt2),\\\\qquad K=\\\\mathbb{Q} <\/p>\n

<\/span><\/span>L<\/span><\/span>=<\/span><\/span><\/span><\/span>Q<\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>,<\/span><\/span><\/span>K<\/span><\/span>=<\/span><\/span><\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u591a\u9879\u5f0f：<\/p>\n

<\/p>\n

x<\/p>\n

2<\/p>\n

\u2212<\/p>\n

2<\/p>\n

x^2-2 <\/p>\n

<\/span><\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u7684\u4e24\u4e2a\u6839\u662f：<\/p>\n

<\/p>\n

2<\/p>\n

,<\/p>\n

\u2212<\/p>\n

2<\/p>\n

\\\\sqrt2,\\\\quad-\\\\sqrt2 <\/p>\n

<\/span><\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>\u2212<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u7531\u4e8e：<\/p>\n

<\/p>\n

2<\/p>\n

\u2208<\/p>\n

L<\/p>\n

\\\\sqrt2\\\\in L <\/p>\n

<\/span><\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2208<\/span><\/span><\/span><\/span>L<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5e76\u4e14：<\/p>\n

<\/p>\n

\u2212<\/p>\n

2<\/p>\n

\u2208<\/p>\n

L<\/p>\n

-\\\\sqrt2\\\\in L <\/p>\n

<\/span><\/span>\u2212<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2208<\/span><\/span><\/span><\/span>L<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6240\u4ee5\u8fd9\u4e2a\u591a\u9879\u5f0f\u7684\u6240\u6709\u6839\u90fd\u5728 <\/p>\n

L<\/p>\n

L<\/p>\n

<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span> \u4e2d\u3002<\/p>\n

\u56e0\u6b64：<\/p>\n

<\/p>\n

Q<\/p>\n

(<\/p>\n

2<\/p>\n

)<\/p>\n

\/<\/p>\n

Q<\/p>\n

\u00a0\u662f\u6b63\u89c4\u6269\u5f20<\/p>\n

\\\\boxed{ \\\\mathbb{Q}(\\\\sqrt2)\/\\\\mathbb{Q} \\\\text{ \u662f\u6b63\u89c4\u6269\u5f20} } <\/p>\n

<\/span><\/span><\/span>Q<\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>\/<\/span>Q<\/span>\u00a0<\/span>\u662f\u6b63\u89c4\u6269\u5f20<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u975e\u6b63\u89c4\u6269\u5f20\u7684\u4f8b\u5b50<\/h3>\n

\u8bbe：<\/p>\n

<\/p>\n

K<\/p>\n

=<\/p>\n

Q<\/p>\n

K=\\\\mathbb{Q}<\/p>\n

<\/span><\/span>K<\/span><\/span>=<\/span><\/span><\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span><\/p>\n

<\/p>\n

L<\/p>\n

=<\/p>\n

Q<\/p>\n

(<\/p>\n

2<\/p>\n

3<\/p>\n

)<\/p>\n

L=\\\\mathbb{Q}(\\\\sqrt[3]{2})<\/p>\n

<\/span><\/span>L<\/span><\/span>=<\/span><\/span><\/span><\/span>Q<\/span>(<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/p>\n

\u90a3\u4e48：<\/p>\n

<\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

=<\/p>\n

Q<\/p>\n

(<\/p>\n

2<\/p>\n

3<\/p>\n

)<\/p>\n

\/<\/p>\n

Q<\/p>\n

L\/K=\\\\mathbb{Q}(\\\\sqrt[3]{2})\/\\\\mathbb{Q} <\/p>\n

<\/span><\/span>L<\/span>\/<\/span>K<\/span><\/span>=<\/span><\/span><\/span><\/span>Q<\/span>(<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>\/<\/span>Q<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u8003\u8651\u591a\u9879\u5f0f：<\/p>\n

<\/p>\n

f<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

=<\/p>\n

x<\/p>\n

3<\/p>\n

\u2212<\/p>\n

2<\/p>\n

f(x)=x^3-2 <\/p>\n

<\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5b83\u5728 <\/p>\n

Q<\/p>\n

\\\\mathbb{Q}<\/p>\n

<\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span> \u4e0a\u4e0d\u53ef\u7ea6，\u5e76\u4e14：<\/p>\n

<\/p>\n

2<\/p>\n

3<\/p>\n

\u2208<\/p>\n

L<\/p>\n

\\\\sqrt[3]{2}\\\\in L <\/p>\n

<\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2208<\/span><\/span><\/span><\/span>L<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4f46\u662f <\/p>\n

f<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

f(x)<\/p>\n

<\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span> \u7684\u53e6\u5916\u4e24\u4e2a\u6839\u4e3a：<\/p>\n

<\/p>\n

\u03c9<\/p>\n

2<\/p>\n

3<\/p>\n

,<\/p>\n

\u03c9<\/p>\n

2<\/p>\n

2<\/p>\n

3<\/p>\n

\\\\omega\\\\sqrt[3]{2}, \\\\qquad \\\\omega^2\\\\sqrt[3]{2} <\/p>\n

<\/span><\/span>\u03c9<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>\u03c9<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d：<\/p>\n

<\/p>\n

\u03c9<\/p>\n

=<\/p>\n

e<\/p>\n

2<\/p>\n

\u03c0<\/p>\n

i<\/p>\n

\/<\/p>\n

3<\/p>\n

\\\\omega=e^{2\\\\pi i\/3} <\/p>\n

<\/span><\/span>\u03c9<\/span><\/span>=<\/span><\/span><\/span><\/span>e<\/span><\/span>2<\/span>\u03c0<\/span>i<\/span>\/3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u800c：<\/p>\n

<\/p>\n

\u03c9<\/p>\n

2<\/p>\n

3<\/p>\n

\u2209<\/p>\n

L<\/p>\n

\\\\omega\\\\sqrt[3]{2}\\\\notin L <\/p>\n

<\/span><\/span>\u03c9<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2208<\/span><\/span><\/span>\/<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>L<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6240\u4ee5：<\/p>\n

<\/p>\n

f<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

\u00a0\u6709\u4e00\u4e2a\u6839\u5728\u00a0<\/p>\n

L<\/p>\n

\u00a0\u4e2d，\u4f46\u975e\u6240\u6709\u6839\u90fd\u5728\u00a0<\/p>\n

L<\/p>\n

\u00a0\u4e2d<\/p>\n

\\\\boxed{ f(x)\\\\text{ \u6709\u4e00\u4e2a\u6839\u5728 }L\\\\text{ \u4e2d，\u4f46\u975e\u6240\u6709\u6839\u90fd\u5728 }L\\\\text{ \u4e2d} } <\/p>\n

<\/span><\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span>\u00a0<\/span>\u6709\u4e00\u4e2a\u6839\u5728<\/span>\u00a0<\/span><\/span>L<\/span>\u00a0<\/span>\u4e2d，\u4f46\u975e\u6240\u6709\u6839\u90fd\u5728<\/span>\u00a0<\/span><\/span>L<\/span>\u00a0<\/span>\u4e2d<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u6b64：<\/p>\n

<\/p>\n

Q<\/p>\n

(<\/p>\n

2<\/p>\n

3<\/p>\n

)<\/p>\n

\/<\/p>\n

Q<\/p>\n

\u00a0\u4e0d\u662f\u6b63\u89c4\u6269\u5f20<\/p>\n

\\\\boxed{ \\\\mathbb{Q}(\\\\sqrt[3]{2})\/\\\\mathbb{Q} \\\\text{ \u4e0d\u662f\u6b63\u89c4\u6269\u5f20} } <\/p>\n

<\/span><\/span><\/span>Q<\/span>(<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>\/<\/span>Q<\/span>\u00a0<\/span>\u4e0d\u662f\u6b63\u89c4\u6269\u5f20<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4e5f\u5c31\u662f\u8bf4：<\/p>\n

<\/p>\n

\u6b63\u89c4\u6269\u5f20：\u4e00\u4e2a\u6839\u8fdb\u6765，\u6240\u6709\u6839\u90fd\u8fdb\u6765<\/p>\n

\\\\boxed{ \\\\text{\u6b63\u89c4\u6269\u5f20：\u4e00\u4e2a\u6839\u8fdb\u6765，\u6240\u6709\u6839\u90fd\u8fdb\u6765} } <\/p>\n

<\/span><\/span><\/span>\u6b63\u89c4\u6269\u5f20：\u4e00\u4e2a\u6839\u8fdb\u6765，\u6240\u6709\u6839\u90fd\u8fdb\u6765<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4f3d\u7f57\u74e6\u6269\u5f20<\/h2>\n

\u56e0\u6b64：<\/p>\n

<\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

\u00a0\u662f\u4f3d\u7f57\u74e6\u6269\u5f20
\n \u2005\u200a<\/p>\n

\u27fa
\n \u2005\u200a<\/p>\n

{<\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

\u00a0\u662f\u53ef\u5206\u6269\u5f20<\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

\u00a0\u662f\u6b63\u89c4\u6269\u5f20<\/p>\n

\\\\boxed{ L\/K\\\\text{ \u662f\u4f3d\u7f57\u74e6\u6269\u5f20} \\\\iff \\\\begin{cases} L\/K\\\\text{ \u662f\u53ef\u5206\u6269\u5f20}\\\\\\\\ L\/K\\\\text{ \u662f\u6b63\u89c4\u6269\u5f20} \\\\end{cases} } <\/p>\n

<\/span><\/span><\/span>L<\/span>\/<\/span>K<\/span>\u00a0<\/span>\u662f\u4f3d\u7f57\u74e6\u6269\u5f20<\/span><\/span><\/span><\/span>\u27fa<\/span><\/span><\/span>{<\/span><\/span><\/span>L<\/span>\/<\/span>K<\/span>\u00a0<\/span>\u662f\u53ef\u5206\u6269\u5f20<\/span><\/span><\/span><\/span><\/span>L<\/span>\/<\/span>K<\/span>\u00a0<\/span>\u662f\u6b63\u89c4\u6269\u5f20<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6b64\u65f6，<\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

L\/K<\/p>\n

<\/span><\/span>L<\/span>\/<\/span>K<\/span><\/span><\/span><\/span><\/span> \u7684\u4f3d\u7f57\u74e6\u7fa4\u5b9a\u4e49\u4e3a：<\/p>\n

<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

)<\/p>\n

=<\/p>\n

{<\/p>\n

\u03c3<\/p>\n

\u2208<\/p>\n

Aut<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

)<\/p>\n

\u2223<\/p>\n

\u03c3<\/p>\n

(<\/p>\n

a<\/p>\n

)<\/p>\n

=<\/p>\n

a<\/p>\n

,<\/p>\n

\u00a0<\/p>\n

\u2200<\/p>\n

a<\/p>\n

\u2208<\/p>\n

K<\/p>\n

}<\/p>\n

\\\\boxed{ \\\\operatorname{Gal}(L\/K) =\\\\{\\\\sigma\\\\in\\\\operatorname{Aut}(L) \\\\mid \\\\sigma(a)=a,\\\\ \\\\forall a\\\\in K \\\\} } <\/p>\n

<\/span><\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>K<\/span>)<\/span><\/span>=<\/span><\/span>{<\/span>\u03c3<\/span><\/span>\u2208<\/span><\/span>Aut<\/span><\/span>(<\/span>L<\/span>)<\/span><\/span>\u2223<\/span><\/span>\u03c3<\/span>(<\/span>a<\/span>)<\/span><\/span>=<\/span><\/span>a<\/span>,<\/span>\u00a0<\/span><\/span>\u2200<\/span>a<\/span><\/span>\u2208<\/span><\/span>K<\/span>}<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d：<\/p>\n

<\/p>\n

Aut<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

)<\/p>\n

\\\\operatorname{Aut}(L)<\/p>\n

<\/span><\/span>Aut<\/span><\/span>(<\/span>L<\/span>)<\/span><\/span><\/span><\/span><\/span> \u662f <\/p>\n

L<\/p>\n

L<\/p>\n

<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span> \u7684\u6240\u6709\u81ea\u540c\u6784\u7ec4\u6210\u7684\u7fa4\u3002<\/p>\n

<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

)<\/p>\n

\\\\operatorname{Gal}(L\/K)<\/p>\n

<\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>K<\/span>)<\/span><\/span><\/span><\/span><\/span> \u662f\u5176\u4e2d\u56fa\u5b9a <\/p>\n

K<\/p>\n

K<\/p>\n

<\/span><\/span>K<\/span><\/span><\/span><\/span><\/span> \u4e2d\u6240\u6709\u5143\u7d20\u7684\u81ea\u540c\u6784\u7ec4\u6210\u7684\u5b50\u7fa4\u3002<\/p>\n

\u56e0\u6b64：<\/p>\n

<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

)<\/p>\n

\u2264<\/p>\n

Aut<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

)<\/p>\n

\\\\boxed{ \\\\operatorname{Gal}(L\/K)\\\\leq\\\\operatorname{Aut}(L) } <\/p>\n

<\/span><\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>K<\/span>)<\/span><\/span>\u2264<\/span><\/span>Aut<\/span><\/span>(<\/span>L<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6b63\u89c4\u5b50\u7fa4\u4e0e\u5546\u7fa4<\/h2>\n

\u5982\u679c <\/p>\n

H<\/p>\n

H<\/p>\n

<\/span><\/span>H<\/span><\/span><\/span><\/span><\/span> \u662f <\/p>\n

G<\/p>\n

G<\/p>\n

<\/span><\/span>G<\/span><\/span><\/span><\/span><\/span> \u7684\u6b63\u89c4\u5b50\u7fa4，\u90a3\u4e48\u53ef\u4ee5\u628a <\/p>\n

G<\/p>\n

G<\/p>\n

<\/span><\/span>G<\/span><\/span><\/span><\/span><\/span> \u6309\u7167 <\/p>\n

H<\/p>\n

H<\/p>\n

<\/span><\/span>H<\/span><\/span><\/span><\/span><\/span> \u5206\u6210\u82e5\u5e72\u4e2a\u966a\u96c6：<\/p>\n

<\/p>\n

g<\/p>\n

H<\/p>\n

gH <\/p>\n

<\/span><\/span>g<\/span>H<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6240\u6709\u966a\u96c6\u7ec4\u6210\u4e00\u4e2a\u7fa4，\u79f0\u4e3a\u5546\u7fa4：<\/p>\n

<\/p>\n

G<\/p>\n

\/<\/p>\n

H<\/p>\n

\\\\boxed{ G\/H } <\/p>\n

<\/span><\/span><\/span>G<\/span>\/<\/span>H<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4e2d\u95f4\u57df\u4e0e\u5546\u7fa4\u5b9a\u7406<\/h2>\n

\u8bbe：<\/p>\n

<\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

L\/K <\/p>\n

<\/span><\/span>L<\/span>\/<\/span>K<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u662f\u4f3d\u7f57\u74e6\u6269\u5f20，\u5e76\u4e14：<\/p>\n

<\/p>\n

K<\/p>\n

\u2286<\/p>\n

M<\/p>\n

\u2286<\/p>\n

L<\/p>\n

K\\\\subseteq M\\\\subseteq L <\/p>\n

<\/span><\/span>K<\/span><\/span>\u2286<\/span><\/span><\/span><\/span>M<\/span><\/span>\u2286<\/span><\/span><\/span><\/span>L<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4ee4：<\/p>\n

<\/p>\n

G<\/p>\n

=<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

)<\/p>\n

G=\\\\operatorname{Gal}(L\/K) <\/p>\n

<\/span><\/span>G<\/span><\/span>=<\/span><\/span><\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>K<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4ee5\u53ca：<\/p>\n

<\/p>\n

H<\/p>\n

=<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

M<\/p>\n

)<\/p>\n

H=\\\\operatorname{Gal}(L\/M) <\/p>\n

<\/span><\/span>H<\/span><\/span>=<\/span><\/span><\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>M<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5982\u679c：<\/p>\n

<\/p>\n

M<\/p>\n

\/<\/p>\n

K<\/p>\n

M\/K <\/p>\n

<\/span><\/span>M<\/span>\/<\/span>K<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4e5f\u662f\u4f3d\u7f57\u74e6\u6269\u5f20，\u90a3\u4e48：<\/p>\n

<\/p>\n

H<\/p>\n

\u22b4<\/p>\n

G<\/p>\n

\\\\boxed{ H\\\\trianglelefteq G } <\/p>\n

<\/span><\/span><\/span>H<\/span><\/span>\u22b4<\/span><\/span>G<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u6b64\u53ef\u4ee5\u6784\u9020\u5546\u7fa4：<\/p>\n

<\/p>\n

G<\/p>\n

\/<\/p>\n

H<\/p>\n

\\\\boxed{ G\/H } <\/p>\n

<\/span><\/span><\/span>G<\/span>\/<\/span>H<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5e76\u4e14\u6709：<\/p>\n

<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

M<\/p>\n

\/<\/p>\n

K<\/p>\n

)<\/p>\n

\u2245<\/p>\n

G<\/p>\n

\/<\/p>\n

H<\/p>\n

\\\\boxed{ \\\\operatorname{Gal}(M\/K) \\\\cong G\/H } <\/p>\n

<\/span><\/span><\/span>Gal<\/span><\/span>(<\/span>M<\/span>\/<\/span>K<\/span>)<\/span><\/span>\u2245<\/span><\/span>G<\/span>\/<\/span>H<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4e5f\u5c31\u662f：<\/p>\n

<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

M<\/p>\n

\/<\/p>\n

K<\/p>\n

)<\/p>\n

\u2245<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

)<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

M<\/p>\n

)<\/p>\n

\\\\boxed{ \\\\operatorname{Gal}(M\/K) \\\\cong \\\\frac{\\\\operatorname{Gal}(L\/K)} {\\\\operatorname{Gal}(L\/M)} } <\/p>\n

<\/span><\/span><\/span>Gal<\/span><\/span>(<\/span>M<\/span>\/<\/span>K<\/span>)<\/span><\/span>\u2245<\/span><\/span><\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>M<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>K<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u6b64：<\/p>\n

<\/p>\n

\u4e2d\u95f4\u57df<\/p>\n

\u27f7<\/p>\n

\u6b63\u89c4\u5b50\u7fa4<\/p>\n

\u27f6<\/p>\n

\u5546\u7fa4<\/p>\n

\\\\boxed{ \\\\text{\u4e2d\u95f4\u57df} \\\\quad\\\\longleftrightarrow\\\\quad \\\\text{\u6b63\u89c4\u5b50\u7fa4} \\\\quad\\\\longrightarrow\\\\quad \\\\text{\u5546\u7fa4} } <\/p>\n

<\/span><\/span><\/span>\u4e2d\u95f4\u57df<\/span><\/span><\/span><\/span>\u27f7<\/span><\/span><\/span>\u6b63\u89c4\u5b50\u7fa4<\/span><\/span><\/span><\/span>\u27f6<\/span><\/span><\/span>\u5546\u7fa4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4f3d\u7f57\u74e6\u5bf9\u5e94\u57fa\u672c\u5b9a\u7406<\/h2>\n

\u5982\u679c <\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

L\/K<\/p>\n

<\/span><\/span>L<\/span>\/<\/span>K<\/span><\/span><\/span><\/span><\/span> \u662f\u4f3d\u7f57\u74e6\u6269\u5f20，M\u662fK\u6269\u5230L\u7684\u4e2d\u95f4\u57df,<\/p>\n

<\/p>\n

K<\/p>\n

\u2286<\/p>\n

M<\/p>\n

\u2286<\/p>\n

L<\/p>\n

K\\\\subseteq M\\\\subseteq L<\/p>\n

<\/span><\/span>K<\/span><\/span>\u2286<\/span><\/span><\/span><\/span>M<\/span><\/span>\u2286<\/span><\/span><\/span><\/span>L<\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4ee4：<\/p>\n

<\/p>\n

G<\/p>\n

=<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

)<\/p>\n

G=\\\\operatorname{Gal}(L\/K)<\/p>\n

<\/span><\/span>G<\/span><\/span>=<\/span><\/span><\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>K<\/span>)<\/span><\/span><\/span><\/span><\/span><\/p>\n

<\/p>\n

H<\/p>\n

=<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

M<\/p>\n

)<\/p>\n

H=\\\\operatorname{Gal}(L\/M)<\/p>\n

<\/span><\/span>H<\/span><\/span>=<\/span><\/span><\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>M<\/span>)<\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5982\u679c <\/p>\n

M<\/p>\n

\/<\/p>\n

K<\/p>\n

M\/K<\/p>\n

<\/span><\/span>M<\/span>\/<\/span>K<\/span><\/span><\/span><\/span><\/span> \u4e5f\u662f\u4f3d\u7f57\u74e6\u6269\u5f20，\u90a3\u4e48：<\/p>\n

<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

M<\/p>\n

\/<\/p>\n

K<\/p>\n

)<\/p>\n

\u2245<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

)<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

M<\/p>\n

)<\/p>\n

\\\\boxed{ \\\\operatorname{Gal}(M\/K) \\\\cong \\\\frac{\\\\operatorname{Gal}(L\/K)} {\\\\operatorname{Gal}(L\/M)} } <\/p>\n

<\/span><\/span><\/span>Gal<\/span><\/span>(<\/span>M<\/span>\/<\/span>K<\/span>)<\/span><\/span>\u2245<\/span><\/span><\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>M<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>K<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

M<\/p>\n

\/<\/p>\n

K<\/p>\n

)<\/p>\n

\u2245<\/p>\n

G<\/p>\n

\/<\/p>\n

H<\/p>\n

\\\\boxed{ \\\\operatorname{Gal}(M\/K)\\\\cong G\/H } <\/p>\n

<\/span><\/span><\/span>Gal<\/span><\/span>(<\/span>M<\/span>\/<\/span>K<\/span>)<\/span><\/span>\u2245<\/span><\/span>G<\/span>\/<\/span>H<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4e94\u6b21\u65b9\u7a0b\u95ee\u9898<\/h2>\n

\u4e00\u822c\u7684\u4e94\u6b21\u65b9\u7a0b：<\/p>\n

<\/p>\n

a<\/p>\n

x<\/p>\n

5<\/p>\n

+<\/p>\n

b<\/p>\n

x<\/p>\n

4<\/p>\n

+<\/p>\n

c<\/p>\n

x<\/p>\n

3<\/p>\n

+<\/p>\n

d<\/p>\n

x<\/p>\n

2<\/p>\n

+<\/p>\n

e<\/p>\n

x<\/p>\n

+<\/p>\n

f<\/p>\n

=<\/p>\n

0<\/p>\n

ax^5+bx^4+cx^3+dx^2+ex+f=0 <\/p>\n

<\/span><\/span>a<\/span>x<\/span><\/span>5<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>b<\/span>x<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>c<\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>d<\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>e<\/span>x<\/span><\/span>+<\/span><\/span><\/span><\/span>f<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4e0e\u4e8c\u6b21\u3001\u4e09\u6b21\u3001\u56db\u6b21\u65b9\u7a0b\u4e0d\u540c，\u4e00\u822c\u4e94\u6b21\u65b9\u7a0b\u4e0d\u5b58\u5728\u53ea\u4f7f\u7528\u52a0\u3001\u51cf\u3001\u4e58\u3001\u9664\u548c\u5f00\u65b9\u8868\u793a\u6839\u7684\u901a\u7528\u516c\u5f0f\u3002<\/p>\n

\u4e5f\u5c31\u662f\u8bf4，\u4e0d\u5b58\u5728\u7c7b\u4f3c\u4e8c\u6b21\u516c\u5f0f：<\/p>\n

<\/p>\n

x<\/p>\n

=<\/p>\n

\u2212<\/p>\n

b<\/p>\n

\u00b1<\/p>\n

b<\/p>\n

2<\/p>\n

\u2212<\/p>\n

4<\/p>\n

a<\/p>\n

c<\/p>\n

2<\/p>\n

a<\/p>\n

x=\\\\frac{-b\\\\pm\\\\sqrt{b^2-4ac}}{2a} <\/p>\n

<\/span><\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>a<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>b<\/span><\/span>\u00b1<\/span><\/span><\/span>b<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>4<\/span>a<\/span>c<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u8fd9\u6837\u7684\u4e94\u6b21\u65b9\u7a0b\u901a\u7528\u6839\u5f0f\u516c\u5f0f\u3002<\/p>\n

\u4f3d\u7f57\u74e6\u7406\u8bba\u7684\u89e3\u91ca<\/h3>\n

\u8bbe <\/p>\n

f<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

\u2208<\/p>\n

K<\/p>\n

[<\/p>\n

x<\/p>\n

]<\/p>\n

f(x)\\\\in K[x]<\/p>\n

<\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span>\u2208<\/span><\/span><\/span><\/span>K<\/span>[<\/span>x<\/span>]<\/span><\/span><\/span><\/span><\/span> \u662f\u4e00\u4e2a\u4e94\u6b21\u591a\u9879\u5f0f，<\/p>\n

L<\/p>\n

L<\/p>\n

<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span> \u662f <\/p>\n

f<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

f(x)<\/p>\n

<\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span> \u7684\u5206\u88c2\u57df\u3002<\/p>\n

\u8003\u8651：<\/p>\n

<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

)<\/p>\n

\\\\operatorname{Gal}(L\/K) <\/p>\n

<\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>K<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5373 <\/p>\n

f<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

f(x)<\/p>\n

<\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span> \u7684\u4f3d\u7f57\u74e6\u7fa4\u3002<\/p>\n

\u4f3d\u7f57\u74e6\u7406\u8bba\u544a\u8bc9\u6211\u4eec：<\/p>\n

<\/p>\n

f<\/p>\n

(<\/p>\n

x<\/p>\n

)<\/p>\n

\u00a0\u53ef\u4ee5\u7528\u6839\u5f0f\u6c42\u89e3
\n \u2005\u200a<\/p>\n

\u27fa
\n \u2005\u200a<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

)<\/p>\n

\u00a0\u662f\u53ef\u89e3\u7fa4<\/p>\n

\\\\boxed{ f(x)\\\\text{ \u53ef\u4ee5\u7528\u6839\u5f0f\u6c42\u89e3} \\\\iff \\\\operatorname{Gal}(L\/K)\\\\text{ \u662f\u53ef\u89e3\u7fa4} } <\/p>\n

<\/span><\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span>\u00a0<\/span>\u53ef\u4ee5\u7528\u6839\u5f0f\u6c42\u89e3<\/span><\/span><\/span><\/span>\u27fa<\/span><\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>K<\/span>)<\/span>\u00a0<\/span>\u662f\u53ef\u89e3\u7fa4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6c42\u6839\u516c\u5f0f\u548c\u6839\u5f0f\u6269\u57df<\/h2>\n

\u5728\u6c42\u6839\u516c\u5f0f\u4e2d，\u6211\u4eec\u4f1a\u4e0d\u65ad\u4f7f\u7528：<\/p>\n

<\/p>\n

+<\/p>\n

,<\/p>\n

\u2212<\/p>\n

,<\/p>\n

\u00d7<\/p>\n

,<\/p>\n

\u00f7<\/p>\n

,<\/p>\n

x<\/p>\n

n<\/p>\n

+,\\\\quad -,\\\\quad \\\\times,\\\\quad \\\\div,\\\\quad \\\\sqrt[n]{\\\\phantom{x}} <\/p>\n

<\/span><\/span>+<\/span>,<\/span><\/span><\/span>\u2212<\/span>,<\/span><\/span><\/span>\u00d7<\/span>,<\/span><\/span><\/span>\u00f7<\/span>,<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d\u771f\u6b63\u4ea7\u751f\u57df\u6269\u5f20\u7684\u662f\u5f00\u65b9\u8fd0\u7b97\u3002<\/p>\n

\u4f8b\u5982：<\/p>\n

<\/p>\n

2<\/p>\n

\\\\sqrt2 <\/p>\n

<\/span><\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u53ef\u4ee5\u770b\u6210\u4ece：<\/p>\n

<\/p>\n

Q<\/p>\n

\\\\mathbb Q <\/p>\n

<\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6269\u5f20\u5230：<\/p>\n

<\/p>\n

Q<\/p>\n

(<\/p>\n

2<\/p>\n

)<\/p>\n

\\\\mathbb Q(\\\\sqrt2) <\/p>\n

<\/span><\/span>Q<\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5373：<\/p>\n

<\/p>\n

Q<\/p>\n

\u2286<\/p>\n

Q<\/p>\n

(<\/p>\n

2<\/p>\n

)<\/p>\n

\\\\boxed{ \\\\mathbb Q \\\\subseteq \\\\mathbb Q(\\\\sqrt2) } <\/p>\n

<\/span><\/span><\/span>Q<\/span><\/span>\u2286<\/span><\/span>Q<\/span>(<\/span><\/span>2<\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\n

\u4e00\u6b21\u5f00\u65b9\u5bf9\u5e94\u4e00\u6b21\u6839\u5f0f\u6269\u57df<\/h3>\n

\u8bbe <\/p>\n

K<\/p>\n

K<\/p>\n

<\/span><\/span>K<\/span><\/span><\/span><\/span><\/span> \u662f\u4e00\u4e2a\u57df，\u52a0\u5165\u4e00\u4e2a <\/p>\n

n<\/p>\n

n<\/p>\n

<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span> \u6b21\u65b9\u6839 <\/p>\n

\u03b1<\/p>\n

\\\\alpha<\/p>\n

<\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span>：<\/p>\n

<\/p>\n

\u03b1<\/p>\n

n<\/p>\n

=<\/p>\n

a<\/p>\n

,<\/p>\n

a<\/p>\n

\u2208<\/p>\n

K<\/p>\n

\\\\alpha^n=a,\\\\qquad a\\\\in K <\/p>\n

<\/span><\/span>\u03b1<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>a<\/span>,<\/span><\/span><\/span>a<\/span><\/span>\u2208<\/span><\/span><\/span><\/span>K<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u90a3\u4e48\u5f97\u5230\u6269\u57df：<\/p>\n

<\/p>\n

K<\/p>\n

\u2286<\/p>\n

K<\/p>\n

(<\/p>\n

\u03b1<\/p>\n

)<\/p>\n

K\\\\subseteq K(\\\\alpha) <\/p>\n

<\/span><\/span>K<\/span><\/span>\u2286<\/span><\/span><\/span><\/span>K<\/span>(<\/span>\u03b1<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5e76\u4e14：<\/p>\n

<\/p>\n

\u03b1<\/p>\n

n<\/p>\n

\u2208<\/p>\n

K<\/p>\n

\\\\boxed{ \\\\alpha^n\\\\in K } <\/p>\n

<\/span><\/span><\/span>\u03b1<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2208<\/span><\/span>K<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u8fd9\u79cd\u6269\u57df\u53eb\u6839\u5f0f\u6269\u57df\u3002<\/p>\n

\u4f8b\u5982：<\/p>\n

<\/p>\n

\u03b1<\/p>\n

=<\/p>\n

2<\/p>\n

3<\/p>\n

\\\\alpha=\\\\sqrt[3]{2} <\/p>\n

<\/span><\/span>\u03b1<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5219：<\/p>\n

<\/p>\n

\u03b1<\/p>\n

3<\/p>\n

=<\/p>\n

2<\/p>\n

\u2208<\/p>\n

Q<\/p>\n

\\\\alpha^3=2\\\\in\\\\mathbb Q <\/p>\n

<\/span><\/span>\u03b1<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>2<\/span><\/span>\u2208<\/span><\/span><\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6240\u4ee5：<\/p>\n

<\/p>\n

Q<\/p>\n

\u2286<\/p>\n

Q<\/p>\n

(<\/p>\n

2<\/p>\n

3<\/p>\n

)<\/p>\n

\\\\boxed{ \\\\mathbb Q\\\\subseteq\\\\mathbb Q(\\\\sqrt[3]{2}) } <\/p>\n

<\/span><\/span><\/span>Q<\/span><\/span>\u2286<\/span><\/span>Q<\/span>(<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\n

\u591a\u6b21\u5f00\u65b9\u5bf9\u5e94\u6839\u5f0f\u6269\u57df\u94fe<\/h3>\n

\u6c42\u6839\u516c\u5f0f\u901a\u5e38\u4e0d\u662f\u53ea\u5f00\u4e00\u6b21\u65b9，\u800c\u662f\u8fde\u7eed\u8fdb\u884c\u591a\u6b21\u5f00\u65b9\u3002<\/p>\n

\u56e0\u6b64\u4f1a\u5f97\u5230\u4e00\u6761\u6269\u57df\u94fe：<\/p>\n

<\/p>\n

K<\/p>\n

=<\/p>\n

K<\/p>\n

0<\/p>\n

\u2286<\/p>\n

K<\/p>\n

1<\/p>\n

\u2286<\/p>\n

K<\/p>\n

2<\/p>\n

\u2286<\/p>\n

\u22ef<\/p>\n

\u2286<\/p>\n

K<\/p>\n

n<\/p>\n

=<\/p>\n

L<\/p>\n

K=K_0 \\\\subseteq K_1 \\\\subseteq K_2 \\\\subseteq\\\\cdots \\\\subseteq K_n=L <\/p>\n

<\/span><\/span>K<\/span><\/span>=<\/span><\/span><\/span><\/span>K<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2286<\/span><\/span><\/span><\/span>K<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2286<\/span><\/span><\/span><\/span>K<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2286<\/span><\/span><\/span><\/span>\u22ef<\/span><\/span>\u2286<\/span><\/span><\/span><\/span>K<\/span><\/span>n<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>L<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d\u6bcf\u4e00\u6b65\u90fd\u662f\u52a0\u5165\u67d0\u4e2a\u6839\u5f0f：<\/p>\n

<\/p>\n

K<\/p>\n

i<\/p>\n

+<\/p>\n

1<\/p>\n

=<\/p>\n

K<\/p>\n

i<\/p>\n

(<\/p>\n

\u03b1<\/p>\n

i<\/p>\n

)<\/p>\n

K_{i+1}=K_i(\\\\alpha_i) <\/p>\n

<\/span><\/span>K<\/span><\/span>i<\/span>+<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>K<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b1<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5e76\u4e14\u5b58\u5728\u67d0\u4e2a\u6b63\u6574\u6570 <\/p>\n

n<\/p>\n

i<\/p>\n

n_i<\/p>\n

<\/span><\/span>n<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u4f7f\u5f97：<\/p>\n

<\/p>\n

\u03b1<\/p>\n

i<\/p>\n

\u2009<\/p>\n

n<\/p>\n

i<\/p>\n

\u2208<\/p>\n

K<\/p>\n

i<\/p>\n

\\\\alpha_i^{\\\\,n_i}\\\\in K_i <\/p>\n

<\/span><\/span>\u03b1<\/span><\/span>i<\/span><\/span><\/span><\/span><\/span>n<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2208<\/span><\/span><\/span><\/span>K<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6240\u4ee5：<\/p>\n

<\/p>\n

K<\/p>\n

i<\/p>\n

+<\/p>\n

1<\/p>\n

=<\/p>\n

K<\/p>\n

i<\/p>\n

(<\/p>\n

\u03b1<\/p>\n

i<\/p>\n

)<\/p>\n

,<\/p>\n

\u03b1<\/p>\n

i<\/p>\n

\u2009<\/p>\n

n<\/p>\n

i<\/p>\n

\u2208<\/p>\n

K<\/p>\n

i<\/p>\n

\\\\boxed{ K_{i+1}=K_i(\\\\alpha_i), \\\\qquad \\\\alpha_i^{\\\\,n_i}\\\\in K_i } <\/p>\n

<\/span><\/span><\/span>K<\/span><\/span>i<\/span>+<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>K<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b1<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>,<\/span><\/span><\/span>\u03b1<\/span><\/span>i<\/span><\/span><\/span><\/span><\/span>n<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2208<\/span><\/span>K<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u8fd9\u6837\u7684\u6269\u57df\u94fe\u79f0\u4e3a\u6839\u5f0f\u6269\u57df\u3002<\/p>\n

\u6839\u5f0f\u6269\u57df\u4f8b\u5b50<\/h3>\n

<\/p>\n

\u03b1<\/p>\n

=<\/p>\n

3<\/p>\n

+<\/p>\n

2<\/p>\n

+<\/p>\n

5<\/p>\n

3<\/p>\n

\\\\alpha=\\\\sqrt{3+\\\\sqrt{2+\\\\sqrt[3]{5}}} <\/p>\n

<\/span><\/span>\u03b1<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>3<\/span><\/span>+<\/span><\/span><\/span>2<\/span><\/span>+<\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>5<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n\n\u6b65\u9aa4\u52a0\u5165\u7684\u6839\u5f0f\u6269\u57df<\/tr>\n\n\n\n\n\n
0<\/td>\n\u2014<\/td>\n<\/p>\n

K<\/p>\n

0<\/p>\n

=<\/p>\n

Q<\/p>\n

K_0=\\\\mathbb Q<\/p>\n

<\/span><\/span>K<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n

1<\/td>\n<\/p>\n

\u03b2<\/p>\n

=<\/p>\n

5<\/p>\n

3<\/p>\n

\\\\beta=\\\\sqrt[3]{5}<\/p>\n

<\/span><\/span>\u03b2<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>5<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n

<\/p>\n

K<\/p>\n

1<\/p>\n

=<\/p>\n

Q<\/p>\n

(<\/p>\n

\u03b2<\/p>\n

)<\/p>\n

K_1=\\\\mathbb Q(\\\\beta)<\/p>\n

<\/span><\/span>K<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>Q<\/span>(<\/span>\u03b2<\/span>)<\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n

2<\/td>\n<\/p>\n

\u03b4<\/p>\n

=<\/p>\n

2<\/p>\n

+<\/p>\n

\u03b2<\/p>\n

\\\\delta=\\\\sqrt{2+\\\\beta}<\/p>\n

<\/span><\/span>\u03b4<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>2<\/span><\/span>+<\/span><\/span>\u03b2<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n

<\/p>\n

K<\/p>\n

2<\/p>\n

=<\/p>\n

K<\/p>\n

1<\/p>\n

(<\/p>\n

\u03b4<\/p>\n

)<\/p>\n

K_2=K_1(\\\\delta)<\/p>\n

<\/span><\/span>K<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>K<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b4<\/span>)<\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n

3<\/td>\n<\/p>\n

\u03b1<\/p>\n

=<\/p>\n

3<\/p>\n

+<\/p>\n

\u03b4<\/p>\n

\\\\alpha=\\\\sqrt{3+\\\\delta}<\/p>\n

<\/span><\/span>\u03b1<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>3<\/span><\/span>+<\/span><\/span>\u03b4<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n

<\/p>\n

K<\/p>\n

3<\/p>\n

=<\/p>\n

K<\/p>\n

2<\/p>\n

(<\/p>\n

\u03b1<\/p>\n

)<\/p>\n

K_3=K_2(\\\\alpha)<\/p>\n

<\/span><\/span>K<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>K<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b1<\/span>)<\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\u56e0\u6b64\u6269\u57df\u8fc7\u7a0b\u4e3a：<\/p>\n

<\/p>\n

Q<\/p>\n

\u2286<\/p>\n

Q<\/p>\n

(<\/p>\n

5<\/p>\n

3<\/p>\n

)<\/p>\n

\u2286<\/p>\n

Q<\/p>\n

(<\/p>\n

5<\/p>\n

3<\/p>\n

,<\/p>\n

2<\/p>\n

+<\/p>\n

5<\/p>\n

3<\/p>\n

)<\/p>\n

\u2286<\/p>\n

Q<\/p>\n

(<\/p>\n

5<\/p>\n

3<\/p>\n

,<\/p>\n

2<\/p>\n

+<\/p>\n

5<\/p>\n

3<\/p>\n

,<\/p>\n

3<\/p>\n

+<\/p>\n

2<\/p>\n

+<\/p>\n

5<\/p>\n

3<\/p>\n

)<\/p>\n

\\\\boxed{ \\\\mathbb Q \\\\subseteq \\\\mathbb Q(\\\\sqrt[3]{5}) \\\\subseteq \\\\mathbb Q(\\\\sqrt[3]{5},\\\\sqrt{2+\\\\sqrt[3]{5}}) \\\\subseteq \\\\mathbb Q(\\\\sqrt[3]{5},\\\\sqrt{2+\\\\sqrt[3]{5}}, \\\\sqrt{3+\\\\sqrt{2+\\\\sqrt[3]{5}}}) } <\/p>\n

<\/span><\/span><\/span>Q<\/span><\/span>\u2286<\/span><\/span>Q<\/span>(<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>5<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>\u2286<\/span><\/span>Q<\/span>(<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>5<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>2<\/span><\/span>+<\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>5<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>\u2286<\/span><\/span>Q<\/span>(<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>5<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>2<\/span><\/span>+<\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>5<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>3<\/span><\/span>+<\/span><\/span><\/span>2<\/span><\/span>+<\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>5<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5373：<\/p>\n

<\/p>\n

\u590d\u6742\u6839\u5f0f<\/p>\n

\u27f7<\/p>\n

\u9010\u5c42\u6839\u5f0f\u6269\u57df<\/p>\n

\\\\boxed{ \\\\text{\u590d\u6742\u6839\u5f0f} \\\\quad\\\\longleftrightarrow\\\\quad \\\\text{\u9010\u5c42\u6839\u5f0f\u6269\u57df} } <\/p>\n

<\/span><\/span><\/span>\u590d\u6742\u6839\u5f0f<\/span><\/span><\/span><\/span>\u27f7<\/span><\/span><\/span>\u9010\u5c42\u6839\u5f0f\u6269\u57df<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5408\u6570\u6b21\u6269\u57df\u53ef\u4ee5\u5206\u6210\u591a\u4e2a\u7d20\u6570\u6b21\u6269\u57df<\/h3>\n

\u8bbe：<\/p>\n

<\/p>\n

[<\/p>\n

K<\/p>\n

(<\/p>\n

\u03b1<\/p>\n

)<\/p>\n

:<\/p>\n

K<\/p>\n

]<\/p>\n

=<\/p>\n

n<\/p>\n

[K(\\\\alpha):K]=n <\/p>\n

<\/span><\/span>[<\/span>K<\/span>(<\/span>\u03b1<\/span>)<\/span><\/span>:<\/span><\/span><\/span><\/span>K<\/span>]<\/span><\/span>=<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5982\u679c <\/p>\n

n<\/p>\n

n<\/p>\n

<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span> \u662f\u5408\u6570，\u4f8b\u5982：<\/p>\n

<\/p>\n

n<\/p>\n

=<\/p>\n

6<\/p>\n

n=6 <\/p>\n

<\/span><\/span>n<\/span><\/span>=<\/span><\/span><\/span><\/span>6<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u90a3\u4e48\u8fd9\u4e2a\u6269\u57df\u53ef\u4ee5\u901a\u8fc7\u4e2d\u95f4\u57df\u62c6\u6210：<\/p>\n

<\/p>\n

K<\/p>\n

\u2286<\/p>\n

M<\/p>\n

\u2286<\/p>\n

K<\/p>\n

(<\/p>\n

\u03b1<\/p>\n

)<\/p>\n

\\\\boxed{ K\\\\subseteq M\\\\subseteq K(\\\\alpha) } <\/p>\n

<\/span><\/span><\/span>K<\/span><\/span>\u2286<\/span><\/span>M<\/span><\/span>\u2286<\/span><\/span>K<\/span>(<\/span>\u03b1<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4f7f\u5f97：<\/p>\n

<\/p>\n

[<\/p>\n

K<\/p>\n

(<\/p>\n

\u03b1<\/p>\n

)<\/p>\n

:<\/p>\n

M<\/p>\n

]<\/p>\n

=<\/p>\n

2<\/p>\n

[K(\\\\alpha):M]=2 <\/p>\n

<\/span><\/span>[<\/span>K<\/span>(<\/span>\u03b1<\/span>)<\/span><\/span>:<\/span><\/span><\/span><\/span>M<\/span>]<\/span><\/span>=<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

<\/p>\n

[<\/p>\n

M<\/p>\n

:<\/p>\n

K<\/p>\n

]<\/p>\n

=<\/p>\n

3<\/p>\n

[M:K]=3 <\/p>\n

<\/span><\/span>[<\/span>M<\/span><\/span>:<\/span><\/span><\/span><\/span>K<\/span>]<\/span><\/span>=<\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6839\u636e\u6269\u57df\u6b21\u6570\u516c\u5f0f：<\/p>\n

<\/p>\n

[<\/p>\n

K<\/p>\n

(<\/p>\n

\u03b1<\/p>\n

)<\/p>\n

:<\/p>\n

K<\/p>\n

]<\/p>\n

=<\/p>\n

[<\/p>\n

K<\/p>\n

(<\/p>\n

\u03b1<\/p>\n

)<\/p>\n

:<\/p>\n

M<\/p>\n

]<\/p>\n

[<\/p>\n

M<\/p>\n

:<\/p>\n

K<\/p>\n

]<\/p>\n

[K(\\\\alpha):K] =[K(\\\\alpha):M][M:K] <\/p>\n

<\/span><\/span>[<\/span>K<\/span>(<\/span>\u03b1<\/span>)<\/span><\/span>:<\/span><\/span><\/span><\/span>K<\/span>]<\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span>K<\/span>(<\/span>\u03b1<\/span>)<\/span><\/span>:<\/span><\/span><\/span><\/span>M<\/span>]<\/span>[<\/span>M<\/span><\/span>:<\/span><\/span><\/span><\/span>K<\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6240\u4ee5：<\/p>\n

<\/p>\n

6<\/p>\n

=<\/p>\n

2<\/p>\n

\u00d7<\/p>\n

3<\/p>\n

6=2\\\\times3 <\/p>\n

<\/span><\/span>6<\/span><\/span>=<\/span><\/span><\/span><\/span>2<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5373：<\/p>\n

<\/p>\n

[<\/p>\n

K<\/p>\n

(<\/p>\n

\u03b1<\/p>\n

)<\/p>\n

:<\/p>\n

K<\/p>\n

]<\/p>\n

=<\/p>\n

6<\/p>\n

\u27f9<\/p>\n

K<\/p>\n

\u2286<\/p>\n

M<\/p>\n

\u2286<\/p>\n

K<\/p>\n

(<\/p>\n

\u03b1<\/p>\n

)<\/p>\n

\\\\boxed{ [K(\\\\alpha):K]=6 \\\\quad\\\\Longrightarrow\\\\quad K\\\\subseteq M\\\\subseteq K(\\\\alpha) } <\/p>\n

<\/span><\/span><\/span>[<\/span>K<\/span>(<\/span>\u03b1<\/span>)<\/span><\/span>:<\/span><\/span>K<\/span>]<\/span><\/span>=<\/span><\/span>6<\/span><\/span><\/span>\u27f9<\/span><\/span><\/span>K<\/span><\/span>\u2286<\/span><\/span>M<\/span><\/span>\u2286<\/span><\/span>K<\/span>(<\/span>\u03b1<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d：<\/p>\n

<\/p>\n

[<\/p>\n

K<\/p>\n

(<\/p>\n

\u03b1<\/p>\n

)<\/p>\n

:<\/p>\n

M<\/p>\n

]<\/p>\n

=<\/p>\n

2<\/p>\n

,<\/p>\n

[<\/p>\n

M<\/p>\n

:<\/p>\n

K<\/p>\n

]<\/p>\n

=<\/p>\n

3<\/p>\n

[K(\\\\alpha):M]=2,\\\\qquad [M:K]=3 <\/p>\n

<\/span><\/span>[<\/span>K<\/span>(<\/span>\u03b1<\/span>)<\/span><\/span>:<\/span><\/span><\/span><\/span>M<\/span>]<\/span><\/span>=<\/span><\/span><\/span><\/span>2<\/span>,<\/span><\/span><\/span>[<\/span>M<\/span><\/span>:<\/span><\/span><\/span><\/span>K<\/span>]<\/span><\/span>=<\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u6b64：<\/p>\n\n\u6269\u57df\u6269\u57df\u6b21\u6570<\/tr>\n\n\n\n\n
<\/p>\n

K<\/p>\n

\u2286<\/p>\n

M<\/p>\n

K\\\\subseteq M<\/p>\n

<\/span><\/span>K<\/span><\/span>\u2286<\/span><\/span><\/span><\/span>M<\/span><\/span><\/span><\/span><\/span><\/td>\n

<\/p>\n

3<\/p>\n

3<\/p>\n

<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n

<\/p>\n

M<\/p>\n

\u2286<\/p>\n

K<\/p>\n

(<\/p>\n

\u03b1<\/p>\n

)<\/p>\n

M\\\\subseteq K(\\\\alpha)<\/p>\n

<\/span><\/span>M<\/span><\/span>\u2286<\/span><\/span><\/span><\/span>K<\/span>(<\/span>\u03b1<\/span>)<\/span><\/span><\/span><\/span><\/span><\/td>\n

<\/p>\n

2<\/p>\n

2<\/p>\n

<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n

<\/p>\n

K<\/p>\n

\u2286<\/p>\n

K<\/p>\n

(<\/p>\n

\u03b1<\/p>\n

)<\/p>\n

K\\\\subseteq K(\\\\alpha)<\/p>\n

<\/span><\/span>K<\/span><\/span>\u2286<\/span><\/span><\/span><\/span>K<\/span>(<\/span>\u03b1<\/span>)<\/span><\/span><\/span><\/span><\/span><\/td>\n

<\/p>\n

6<\/p>\n

=<\/p>\n

3<\/p>\n

\u00d7<\/p>\n

2<\/p>\n

6=3\\\\times2<\/p>\n

<\/span><\/span>6<\/span><\/span>=<\/span><\/span><\/span><\/span>3<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\n

\u4e00\u822c\u60c5\u51b5<\/h4>\n

\u5982\u679c：<\/p>\n

<\/p>\n

[<\/p>\n

L<\/p>\n

:<\/p>\n

K<\/p>\n

]<\/p>\n

=<\/p>\n

n<\/p>\n

[L:K]=n <\/p>\n

<\/span><\/span>[<\/span>L<\/span><\/span>:<\/span><\/span><\/span><\/span>K<\/span>]<\/span><\/span>=<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4e14：<\/p>\n

<\/p>\n

n<\/p>\n

=<\/p>\n

p<\/p>\n

1<\/p>\n

p<\/p>\n

2<\/p>\n

\u22ef<\/p>\n

p<\/p>\n

r<\/p>\n

n=p_1p_2\\\\cdots p_r <\/p>\n

<\/span><\/span>n<\/span><\/span>=<\/span><\/span><\/span><\/span>p<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>p<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ef<\/span><\/span>p<\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d\u6bcf\u4e2a <\/p>\n

p<\/p>\n

i<\/p>\n

p_i<\/p>\n

<\/span><\/span>p<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u90fd\u662f\u7d20\u6570，\u90a3\u4e48\u53ef\u4ee5\u8003\u8651\u628a\u6269\u57df\u5206\u89e3\u6210\u82e5\u5e72\u4e2a\u7d20\u6570\u6b21\u6269\u57df：<\/p>\n

<\/p>\n

K<\/p>\n

=<\/p>\n

K<\/p>\n

0<\/p>\n

\u2286<\/p>\n

K<\/p>\n

1<\/p>\n

\u2286<\/p>\n

K<\/p>\n

2<\/p>\n

\u2286<\/p>\n

\u22ef<\/p>\n

\u2286<\/p>\n

K<\/p>\n

r<\/p>\n

=<\/p>\n

L<\/p>\n

\\\\boxed{ K=K_0 \\\\subseteq K_1 \\\\subseteq K_2 \\\\subseteq\\\\cdots \\\\subseteq K_r=L } <\/p>\n

<\/span><\/span><\/span>K<\/span><\/span>=<\/span><\/span>K<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2286<\/span><\/span>K<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2286<\/span><\/span>K<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2286<\/span><\/span>\u22ef<\/span><\/span>\u2286<\/span><\/span>K<\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6ee1\u8db3：<\/p>\n

<\/p>\n

[<\/p>\n

K<\/p>\n

i<\/p>\n

:<\/p>\n

K<\/p>\n

i<\/p>\n

\u2212<\/p>\n

1<\/p>\n

]<\/p>\n

=<\/p>\n

p<\/p>\n

i<\/p>\n

\\\\boxed{ [K_i:K_{i-1}]=p_i } <\/p>\n

<\/span><\/span><\/span>[<\/span>K<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>:<\/span><\/span>K<\/span><\/span>i<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span>=<\/span><\/span>p<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4e8e\u662f：<\/p>\n

<\/p>\n

[<\/p>\n

L<\/p>\n

:<\/p>\n

K<\/p>\n

]<\/p>\n

=<\/p>\n

\u220f<\/p>\n

i<\/p>\n

=<\/p>\n

1<\/p>\n

r<\/p>\n

[<\/p>\n

K<\/p>\n

i<\/p>\n

:<\/p>\n

K<\/p>\n

i<\/p>\n

\u2212<\/p>\n

1<\/p>\n

]<\/p>\n

=<\/p>\n

p<\/p>\n

1<\/p>\n

p<\/p>\n

2<\/p>\n

\u22ef<\/p>\n

p<\/p>\n

r<\/p>\n

[L:K] =\\\\prod_{i=1}^{r} [K_i:K_{i-1}] =p_1p_2\\\\cdots p_r <\/p>\n

<\/span><\/span>[<\/span>L<\/span><\/span>:<\/span><\/span><\/span><\/span>K<\/span>]<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>i<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>\u220f<\/span><\/span><\/span><\/span>r<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>[<\/span>K<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>:<\/span><\/span><\/span><\/span>K<\/span><\/span>i<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span>=<\/span><\/span><\/span><\/span>p<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>p<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ef<\/span><\/span>p<\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u6240\u4ee5： <\/p>\n

\u5408\u6570\u6b21\u6269\u57df<\/p>\n

\u27f6<\/p>\n

\u591a\u4e2a\u7d20\u6570\u6b21\u6269\u57df<\/p>\n

\\\\boxed{ \\\\text{\u5408\u6570\u6b21\u6269\u57df} \\\\quad\\\\longrightarrow\\\\quad \\\\text{\u591a\u4e2a\u7d20\u6570\u6b21\u6269\u57df} } <\/p>\n

<\/span><\/span><\/span>\u5408\u6570\u6b21\u6269\u57df<\/span><\/span><\/span><\/span>\u27f6<\/span><\/span><\/span>\u591a\u4e2a\u7d20\u6570\u6b21\u6269\u57df<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u5f00mn\u6b21\u6839\u53f7 \u53ef\u4ee5\u5206\u4e3a\u5148\u5f00m\u6b21\u6839\u53f7, \u518d\u5f00n\u6b21\u6839\u53f7 <\/p>\n

a<\/p>\n

m<\/p>\n

n<\/p>\n

=<\/p>\n

a<\/p>\n

m<\/p>\n

n<\/p>\n

\\\\sqrt[mn]{a} =\\\\sqrt[n]{\\\\sqrt[m]{a}} <\/p>\n

<\/span><\/span><\/span>mn<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u6b64：<\/p>\n

<\/p>\n

a<\/p>\n

m<\/p>\n

n<\/p>\n

\u27f6<\/p>\n

a<\/p>\n

m<\/p>\n

\u27f6<\/p>\n

a<\/p>\n

m<\/p>\n

n<\/p>\n

\\\\boxed{ \\\\sqrt[mn]{a} \\\\quad\\\\longrightarrow\\\\quad \\\\sqrt[m]{a} \\\\quad\\\\longrightarrow\\\\quad \\\\sqrt[n]{\\\\sqrt[m]{a}} } <\/p>\n

<\/span><\/span><\/span><\/span>mn<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u27f6<\/span><\/span><\/span><\/span>m<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u27f6<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5bf9\u5e94\u6269\u57df：<\/p>\n

<\/p>\n

K<\/p>\n

\u2286<\/p>\n

K<\/p>\n

(<\/p>\n

a<\/p>\n

m<\/p>\n

)<\/p>\n

\u2286<\/p>\n

K<\/p>\n

(<\/p>\n

a<\/p>\n

m<\/p>\n

,<\/p>\n

a<\/p>\n

m<\/p>\n

n<\/p>\n

)<\/p>\n

K \\\\subseteq K(\\\\sqrt[m]{a}) \\\\subseteq K\\\\left(\\\\sqrt[m]{a},\\\\sqrt[n]{\\\\sqrt[m]{a}}\\\\right) <\/p>\n

<\/span><\/span>K<\/span><\/span>\u2286<\/span><\/span><\/span><\/span>K<\/span>(<\/span><\/span>m<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>\u2286<\/span><\/span><\/span><\/span>K<\/span><\/span>(<\/span><\/span><\/span>m<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>m<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4f8b\u5982：<\/p>\n

<\/p>\n

2<\/p>\n

6<\/p>\n

=<\/p>\n

2<\/p>\n

3<\/p>\n

\\\\sqrt[6]{2} =\\\\sqrt{\\\\sqrt[3]{2}} <\/p>\n

<\/span><\/span><\/span>6<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5bf9\u5e94：<\/p>\n

<\/p>\n

Q<\/p>\n

\u2286<\/p>\n

Q<\/p>\n

(<\/p>\n

2<\/p>\n

3<\/p>\n

)<\/p>\n

\u2286<\/p>\n

Q<\/p>\n

(<\/p>\n

2<\/p>\n

3<\/p>\n

,<\/p>\n

2<\/p>\n

3<\/p>\n

)<\/p>\n

\\\\mathbb Q \\\\subseteq \\\\mathbb Q(\\\\sqrt[3]{2}) \\\\subseteq \\\\mathbb Q\\\\left(\\\\sqrt[3]{2},\\\\sqrt{\\\\sqrt[3]{2}}\\\\right) <\/p>\n

<\/span><\/span>Q<\/span><\/span>\u2286<\/span><\/span><\/span><\/span>Q<\/span>(<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>\u2286<\/span><\/span><\/span><\/span>Q<\/span><\/span>(<\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4e00\u5143\u4e8c\u6b21\u6c42\u6839\u516c\u5f0f\u7684\u6269\u57df\u8fc7\u7a0b<\/h2>\n

<\/p>\n

a<\/p>\n

x<\/p>\n

2<\/p>\n

+<\/p>\n

b<\/p>\n

x<\/p>\n

+<\/p>\n

c<\/p>\n

=<\/p>\n

0<\/p>\n

ax^2+bx+c=0 <\/p>\n

<\/span><\/span>a<\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>b<\/span>x<\/span><\/span>+<\/span><\/span><\/span><\/span>c<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n\n\u6b65\u9aa4\u64cd\u4f5c\u6269\u57df<\/tr>\n\n\n\n\n\n
0<\/td>\n\u521d\u59cb<\/td>\n<\/p>\n

K<\/p>\n

K<\/p>\n

<\/span><\/span>K<\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n

1<\/td>\n<\/p>\n

\u0394<\/p>\n

=<\/p>\n

b<\/p>\n

2<\/p>\n

\u2212<\/p>\n

4<\/p>\n

a<\/p>\n

c<\/p>\n

\\\\Delta=b^2-4ac<\/p>\n

<\/span><\/span>\u0394<\/span><\/span>=<\/span><\/span><\/span><\/span>b<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>4<\/span>a<\/span>c<\/span><\/span><\/span><\/span><\/span><\/td>\n

<\/p>\n

K<\/p>\n

K<\/p>\n

<\/span><\/span>K<\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n

2<\/td>\n\u52a0\u5165 <\/p>\n

\u0394<\/p>\n

\\\\sqrt{\\\\Delta}<\/p>\n

<\/span><\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n

<\/p>\n

K<\/p>\n

(<\/p>\n

\u0394<\/p>\n

)<\/p>\n

K(\\\\sqrt{\\\\Delta})<\/p>\n

<\/span><\/span>K<\/span>(<\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n

3<\/td>\n\u5f97\u5230\u6839<\/td>\n<\/p>\n

x<\/p>\n

=<\/p>\n

\u2212<\/p>\n

b<\/p>\n

\u00b1<\/p>\n

\u0394<\/p>\n

2<\/p>\n

a<\/p>\n

x=\\\\frac{-b\\\\pm\\\\sqrt{\\\\Delta}}{2a}<\/p>\n

<\/span><\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>a<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span>b<\/span>\u00b1<\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\u4e00\u5143\u4e09\u6b21\u6c42\u6839\u516c\u5f0f\u7684\u6269\u57df\u8fc7\u7a0b<\/h2>\n

<\/p>\n

M<\/p>\n

\/<\/p>\n

K<\/p>\n

\u00a0\u662f\u6b63\u89c4\u6269\u5f20
\n \u2005\u200a<\/p>\n

\u27fa
\n \u2005\u200a<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

M<\/p>\n

)<\/p>\n

\u22b4<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

)<\/p>\n

\\\\boxed{ M\/K\\\\text{ \u662f\u6b63\u89c4\u6269\u5f20} \\\\iff \\\\operatorname{Gal}(L\/M) \\\\trianglelefteq \\\\operatorname{Gal}(L\/K) } <\/p>\n

<\/span><\/span><\/span>M<\/span>\/<\/span>K<\/span>\u00a0<\/span>\u662f\u6b63\u89c4\u6269\u5f20<\/span><\/span><\/span><\/span>\u27fa<\/span><\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>M<\/span>)<\/span><\/span>\u22b4<\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>K<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d：<\/p>\n

<\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

\u00a0\u662f\u4f3d\u7f57\u74e6\u6269\u5f20<\/p>\n

,<\/p>\n

K<\/p>\n

\u2286<\/p>\n

M<\/p>\n

\u2286<\/p>\n

L<\/p>\n

\\\\boxed{ L\/K\\\\text{ \u662f\u4f3d\u7f57\u74e6\u6269\u5f20}, \\\\qquad K\\\\subseteq M\\\\subseteq L } <\/p>\n

<\/span><\/span><\/span>L<\/span>\/<\/span>K<\/span>\u00a0<\/span>\u662f\u4f3d\u7f57\u74e6\u6269\u5f20<\/span><\/span>,<\/span><\/span><\/span>K<\/span><\/span>\u2286<\/span><\/span>M<\/span><\/span>\u2286<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u8bbe\u901a\u7528\u4e00\u5143\u4e09\u6b21\u65b9\u7a0b：<\/p>\n

<\/p>\n

a<\/p>\n

x<\/p>\n

3<\/p>\n

+<\/p>\n

b<\/p>\n

x<\/p>\n

2<\/p>\n

+<\/p>\n

c<\/p>\n

x<\/p>\n

+<\/p>\n

d<\/p>\n

=<\/p>\n

0<\/p>\n

ax^3+bx^2+cx+d=0 <\/p>\n

<\/span><\/span>a<\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>b<\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>c<\/span>x<\/span><\/span>+<\/span><\/span><\/span><\/span>d<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u7ecf\u8fc7\u53d8\u91cf\u4ee3\u6362\u5316\u4e3a：<\/p>\n

<\/p>\n

y<\/p>\n

3<\/p>\n

+<\/p>\n

p<\/p>\n

y<\/p>\n

+<\/p>\n

q<\/p>\n

=<\/p>\n

0<\/p>\n

y^3+py+q=0 <\/p>\n

<\/span><\/span>y<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>p<\/span>y<\/span><\/span>+<\/span><\/span><\/span><\/span>q<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n\n\u6b65\u9aa4\u64cd\u4f5c\u6269\u57df<\/tr>\n\n\n\n\n\n\n\n
0<\/td>\n\u521d\u59cb<\/td>\n<\/p>\n

K<\/p>\n

K<\/p>\n

<\/span><\/span>K<\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n

1<\/td>\n<\/p>\n

\u0394<\/p>\n

=<\/p>\n

(<\/p>\n

q<\/p>\n

2<\/p>\n

)<\/p>\n

2<\/p>\n

+<\/p>\n

(<\/p>\n

p<\/p>\n

3<\/p>\n

)<\/p>\n

3<\/p>\n

\\\\Delta=\\\\left(\\\\frac q2\\\\right)^2+\\\\left(\\\\frac p3\\\\right)^3<\/p>\n

<\/span><\/span>\u0394<\/span><\/span>=<\/span><\/span><\/span><\/span>(<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>q<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>(<\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>p<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n

<\/p>\n

K<\/p>\n

K<\/p>\n

<\/span><\/span>K<\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n

2<\/td>\n\u52a0\u5165 <\/p>\n

\u0394<\/p>\n

\\\\sqrt{\\\\Delta}<\/p>\n

<\/span><\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n

<\/p>\n

K<\/p>\n

(<\/p>\n

\u0394<\/p>\n

)<\/p>\n

K(\\\\sqrt{\\\\Delta})<\/p>\n

<\/span><\/span>K<\/span>(<\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n

3<\/td>\n\u52a0\u5165 <\/p>\n

\u2212<\/p>\n

q<\/p>\n

2<\/p>\n

+<\/p>\n

\u0394<\/p>\n

3<\/p>\n

\\\\sqrt[3]{-\\\\frac q2+\\\\sqrt{\\\\Delta}}<\/p>\n

<\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>q<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n

<\/p>\n

K<\/p>\n

(<\/p>\n

\u0394<\/p>\n

,<\/p>\n

\u2212<\/p>\n

q<\/p>\n

2<\/p>\n

+<\/p>\n

\u0394<\/p>\n

3<\/p>\n

)<\/p>\n

K(\\\\sqrt{\\\\Delta},\\\\sqrt[3]{-\\\\frac q2+\\\\sqrt{\\\\Delta}})<\/p>\n

<\/span><\/span>K<\/span>(<\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>q<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n

4<\/td>\n\u52a0\u5165 <\/p>\n

\u2212<\/p>\n

q<\/p>\n

2<\/p>\n

\u2212<\/p>\n

\u0394<\/p>\n

3<\/p>\n

\\\\sqrt[3]{-\\\\frac q2-\\\\sqrt{\\\\Delta}}<\/p>\n

<\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>q<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n

<\/p>\n

K<\/p>\n

(<\/p>\n

\u0394<\/p>\n

,<\/p>\n

\u2212<\/p>\n

q<\/p>\n

2<\/p>\n

+<\/p>\n

\u0394<\/p>\n

3<\/p>\n

,<\/p>\n

\u2212<\/p>\n

q<\/p>\n

2<\/p>\n

\u2212<\/p>\n

\u0394<\/p>\n

3<\/p>\n

)<\/p>\n

K(\\\\sqrt{\\\\Delta},\\\\sqrt[3]{-\\\\frac q2+\\\\sqrt{\\\\Delta}},\\\\sqrt[3]{-\\\\frac q2-\\\\sqrt{\\\\Delta}})<\/p>\n

<\/span><\/span>K<\/span>(<\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>q<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>q<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n

5<\/td>\n\u5f97\u5230\u4e09\u4e2a\u6839<\/td>\n<\/p>\n

x<\/p>\n

1<\/p>\n

,<\/p>\n

x<\/p>\n

2<\/p>\n

,<\/p>\n

x<\/p>\n

3<\/p>\n

x_1,x_2,x_3<\/p>\n

<\/span><\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

<\/p>\n

K<\/p>\n

\u2286<\/p>\n

K<\/p>\n

(<\/p>\n

\u0394<\/p>\n

)<\/p>\n

\u2286<\/p>\n

K<\/p>\n

(<\/p>\n

\u0394<\/p>\n

,<\/p>\n

\u2212<\/p>\n

q<\/p>\n

2<\/p>\n

+<\/p>\n

\u0394<\/p>\n

3<\/p>\n

)<\/p>\n

\u2286<\/p>\n

L<\/p>\n

\\\\boxed{ K \\\\subseteq K(\\\\sqrt{\\\\Delta}) \\\\subseteq K\\\\left( \\\\sqrt{\\\\Delta}, \\\\sqrt[3]{-\\\\frac q2+\\\\sqrt{\\\\Delta}} \\\\right) \\\\subseteq L } <\/p>\n

<\/span><\/span><\/span>K<\/span><\/span>\u2286<\/span><\/span>K<\/span>(<\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>\u2286<\/span><\/span>K<\/span><\/span>(<\/span><\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>q<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>\u2286<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u6216 <\/p>\n

K<\/p>\n

\u2286<\/p>\n

K<\/p>\n

(<\/p>\n

\u0394<\/p>\n

)<\/p>\n

\u2286<\/p>\n

L<\/p>\n

\\\\boxed{ K \\\\subseteq K(\\\\sqrt{\\\\Delta}) \\\\subseteq L } <\/p>\n

<\/span><\/span><\/span>K<\/span><\/span>\u2286<\/span><\/span>K<\/span>(<\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>\u2286<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5bf9\u5e94\u7684\u7fa4\u94fe\u4e3a：<\/p>\n

<\/p>\n

S<\/p>\n

3<\/p>\n

\u2287<\/p>\n

A<\/p>\n

3<\/p>\n

\u2287<\/p>\n

{<\/p>\n

e<\/p>\n

}<\/p>\n

\\\\boxed{ S_3 \\\\supseteq A_3 \\\\supseteq \\\\{e\\\\} } <\/p>\n

<\/span><\/span><\/span>S<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2287<\/span><\/span>A<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2287<\/span><\/span>{<\/span>e<\/span>}<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5bf9\u5e94\u5173\u7cfb\u53ef\u4ee5\u5199\u6210：<\/p>\n

<\/p>\n

K<\/p>\n

\u27f7<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

)<\/p>\n

\u2245<\/p>\n

S<\/p>\n

3<\/p>\n

K<\/p>\n

(<\/p>\n

\u0394<\/p>\n

)<\/p>\n

\u27f7<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

(<\/p>\n

\u0394<\/p>\n

)<\/p>\n

)<\/p>\n

\u2245<\/p>\n

A<\/p>\n

3<\/p>\n

L<\/p>\n

\u27f7<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

L<\/p>\n

)<\/p>\n

=<\/p>\n

{<\/p>\n

e<\/p>\n

}<\/p>\n

\\\\boxed{ \\\\begin{array}{ccc} K &\\\\longleftrightarrow& \\\\operatorname{Gal}(L\/K)\\\\cong S_3 \\\\\\\\[6pt] K(\\\\sqrt{\\\\Delta}) &\\\\longleftrightarrow& \\\\operatorname{Gal}(L\/K(\\\\sqrt{\\\\Delta}))\\\\cong A_3 \\\\\\\\[6pt] L &\\\\longleftrightarrow& \\\\operatorname{Gal}(L\/L)=\\\\{e\\\\} \\\\end{array} } <\/p>\n

<\/span><\/span><\/span><\/span><\/span>K<\/span><\/span><\/span><\/span>K<\/span>(<\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>L<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u27f7<\/span><\/span><\/span><\/span>\u27f7<\/span><\/span><\/span><\/span>\u27f7<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>K<\/span>)<\/span><\/span>\u2245<\/span><\/span>S<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>K<\/span>(<\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>))<\/span><\/span>\u2245<\/span><\/span>A<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>L<\/span>)<\/span><\/span>=<\/span><\/span>{<\/span>e<\/span>}<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

<\/p>\n

\u0394<\/p>\n

\\\\sqrt{\\\\Delta}<\/p>\n

<\/span><\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u4e0e <\/p>\n

A<\/p>\n

3<\/p>\n

A_3<\/p>\n

<\/span><\/span>A<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u7684\u5173\u7cfb<\/h4>\n

\u8fd9\u91cc\u6362\u4e2a\u548cCardano \u516c\u5f0f\u7b49\u6548\u7684\u5224\u522b\u5f0f<\/p>\n

\u4e09\u6b21\u65b9\u7a0b\u7684\u4e09\u4e2a\u6839\u4e3a：<\/p>\n

<\/p>\n

x<\/p>\n

1<\/p>\n

,<\/p>\n

x<\/p>\n

2<\/p>\n

,<\/p>\n

x<\/p>\n

3<\/p>\n

x_1,x_2,x_3 <\/p>\n

<\/span><\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5224\u522b\u5f0f\u4e3a：<\/p>\n

<\/p>\n

\u0394<\/p>\n

=<\/p>\n

(<\/p>\n

x<\/p>\n

1<\/p>\n

\u2212<\/p>\n

x<\/p>\n

2<\/p>\n

)<\/p>\n

2<\/p>\n

(<\/p>\n

x<\/p>\n

1<\/p>\n

\u2212<\/p>\n

x<\/p>\n

3<\/p>\n

)<\/p>\n

2<\/p>\n

(<\/p>\n

x<\/p>\n

2<\/p>\n

\u2212<\/p>\n

x<\/p>\n

3<\/p>\n

)<\/p>\n

2<\/p>\n

=<\/p>\n

\u2212<\/p>\n

108<\/p>\n

(<\/p>\n

(<\/p>\n

q<\/p>\n

2<\/p>\n

)<\/p>\n

2<\/p>\n

+<\/p>\n

(<\/p>\n

p<\/p>\n

3<\/p>\n

)<\/p>\n

3<\/p>\n

)<\/p>\n

\\\\boxed{ \\\\Delta= (x_1-x_2)^2 (x_1-x_3)^2 (x_2-x_3)^2} =-108 \\\\left( \\\\left(\\\\frac q2\\\\right)^2+ \\\\left(\\\\frac p3\\\\right)^3 \\\\right) <\/p>\n

<\/span><\/span><\/span>\u0394<\/span><\/span>=<\/span><\/span>(<\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>108<\/span><\/span>(<\/span><\/span>(<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>q<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span>(<\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span>p<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u6b64：<\/p>\n

<\/p>\n

\u0394<\/p>\n

=<\/p>\n

\u00b1<\/p>\n

(<\/p>\n

x<\/p>\n

1<\/p>\n

\u2212<\/p>\n

x<\/p>\n

2<\/p>\n

)<\/p>\n

(<\/p>\n

x<\/p>\n

1<\/p>\n

\u2212<\/p>\n

x<\/p>\n

3<\/p>\n

)<\/p>\n

(<\/p>\n

x<\/p>\n

2<\/p>\n

\u2212<\/p>\n

x<\/p>\n

3<\/p>\n

)<\/p>\n

\\\\boxed{ \\\\sqrt{\\\\Delta} =\\\\pm (x_1-x_2)(x_1-x_3)(x_2-x_3) } <\/p>\n

<\/span><\/span><\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u00b1<\/span>(<\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>(<\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>(<\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5b83\u53ef\u4ee5\u533a\u5206\u6839\u7684\u5076\u7f6e\u6362\u548c\u5947\u7f6e\u6362：<\/p>\n

<\/p>\n

\u5076\u7f6e\u6362<\/p>\n

:<\/p>\n

\u0394<\/p>\n

\u21a6<\/p>\n

\u0394<\/p>\n

\u5947\u7f6e\u6362<\/p>\n

:<\/p>\n

\u0394<\/p>\n

\u21a6<\/p>\n

\u2212<\/p>\n

\u0394<\/p>\n

\\\\boxed{ \\\\begin{aligned} \\\\text{\u5076\u7f6e\u6362}&:\\\\quad \\\\sqrt{\\\\Delta}\\\\mapsto\\\\sqrt{\\\\Delta} \\\\\\\\ \\\\text{\u5947\u7f6e\u6362}&:\\\\quad \\\\sqrt{\\\\Delta}\\\\mapsto-\\\\sqrt{\\\\Delta} \\\\end{aligned} } <\/p>\n

<\/span><\/span><\/span><\/span>\u5076\u7f6e\u6362<\/span><\/span><\/span><\/span><\/span>\u5947\u7f6e\u6362<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>:<\/span><\/span><\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u21a6<\/span><\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>:<\/span><\/span><\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u21a6<\/span><\/span>\u2212<\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56fa\u5b9a <\/p>\n

\u0394<\/p>\n

\\\\sqrt{\\\\Delta}<\/p>\n

<\/span><\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u7684\u81ea\u540c\u6784\u6070\u597d\u7ec4\u6210 <\/p>\n

A<\/p>\n

3<\/p>\n

A_3<\/p>\n

<\/span><\/span>A<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u56e0\u6b64 <\/p>\n

\u0394<\/p>\n

\\\\sqrt{\\\\Delta}<\/p>\n

<\/span><\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u7684\u4e0d\u52a8\u57df\u4e3a <\/p>\n

K<\/p>\n

(<\/p>\n

\u0394<\/p>\n

)<\/p>\n

K(\\\\sqrt{\\\\Delta})<\/p>\n

<\/span><\/span>K<\/span>(<\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002 \u6240\u4ee5： <\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

(<\/p>\n

\u0394<\/p>\n

)<\/p>\n

)<\/p>\n

\u2245<\/p>\n

A<\/p>\n

3<\/p>\n

\\\\boxed{ \\\\operatorname{Gal}(L\/K(\\\\sqrt{\\\\Delta})) \\\\cong A_3 } <\/p>\n

<\/span><\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>K<\/span>(<\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>))<\/span><\/span>\u2245<\/span><\/span>A<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4e00\u5143\u4e94\u6b21\u6c42\u6839\u516c\u5f0f\u7684\u6269\u57df\u8fc7\u7a0b<\/h2>\n

\u8bbe\u4e00\u822c\u4e00\u5143\u4e94\u6b21\u65b9\u7a0b：<\/p>\n

<\/p>\n

a<\/p>\n

x<\/p>\n

5<\/p>\n

+<\/p>\n

b<\/p>\n

x<\/p>\n

4<\/p>\n

+<\/p>\n

c<\/p>\n

x<\/p>\n

3<\/p>\n

+<\/p>\n

d<\/p>\n

x<\/p>\n

2<\/p>\n

+<\/p>\n

e<\/p>\n

x<\/p>\n

+<\/p>\n

f<\/p>\n

=<\/p>\n

0<\/p>\n

ax^5+bx^4+cx^3+dx^2+ex+f=0 <\/p>\n

<\/span><\/span>a<\/span>x<\/span><\/span>5<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>b<\/span>x<\/span><\/span>4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>c<\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>d<\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>e<\/span>x<\/span><\/span>+<\/span><\/span><\/span><\/span>f<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u8bbe\u5b83\u7684\u4e94\u4e2a\u6839\u4e3a：<\/p>\n

<\/p>\n

x<\/p>\n

1<\/p>\n

,<\/p>\n

x<\/p>\n

2<\/p>\n

,<\/p>\n

x<\/p>\n

3<\/p>\n

,<\/p>\n

x<\/p>\n

4<\/p>\n

,<\/p>\n

x<\/p>\n

5<\/p>\n

x_1,x_2,x_3,x_4,x_5 <\/p>\n

<\/span><\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>x<\/span><\/span>4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>x<\/span><\/span>5<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u8bbe <\/p>\n

L<\/p>\n

L<\/p>\n

<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span> \u662f\u8fd9\u4e2a\u65b9\u7a0b\u7684\u5206\u88c2\u57df：<\/p>\n

<\/p>\n

L<\/p>\n

=<\/p>\n

K<\/p>\n

(<\/p>\n

x<\/p>\n

1<\/p>\n

,<\/p>\n

x<\/p>\n

2<\/p>\n

,<\/p>\n

x<\/p>\n

3<\/p>\n

,<\/p>\n

x<\/p>\n

4<\/p>\n

,<\/p>\n

x<\/p>\n

5<\/p>\n

)<\/p>\n

\\\\boxed{ L=K(x_1,x_2,x_3,x_4,x_5) } <\/p>\n

<\/span><\/span><\/span>L<\/span><\/span>=<\/span><\/span>K<\/span>(<\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>x<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>x<\/span><\/span>4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>x<\/span><\/span>5<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5982\u679c\u4e94\u6b21\u65b9\u7a0b\u53ef\u4ee5\u7528\u6839\u5f0f\u6c42\u89e3，\u90a3\u4e48\u53ef\u4ee5\u5efa\u7acb\u6839\u5f0f\u6269\u57df\u94fe：<\/p>\n

<\/p>\n

K<\/p>\n

=<\/p>\n

K<\/p>\n

0<\/p>\n

\u2286<\/p>\n

K<\/p>\n

1<\/p>\n

\u2286<\/p>\n

K<\/p>\n

2<\/p>\n

\u2286<\/p>\n

\u22ef<\/p>\n

\u2286<\/p>\n

K<\/p>\n

n<\/p>\n

=<\/p>\n

L<\/p>\n

\\\\boxed{ K=K_0 \\\\subseteq K_1 \\\\subseteq K_2 \\\\subseteq \\\\cdots \\\\subseteq K_n=L } <\/p>\n

<\/span><\/span><\/span>K<\/span><\/span>=<\/span><\/span>K<\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2286<\/span><\/span>K<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2286<\/span><\/span>K<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2286<\/span><\/span>\u22ef<\/span><\/span>\u2286<\/span><\/span>K<\/span><\/span>n<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d\u6bcf\u4e00\u6b65\u90fd\u662f\u52a0\u5165\u4e00\u4e2a\u6839\u5f0f：<\/p>\n

<\/p>\n

K<\/p>\n

i<\/p>\n

+<\/p>\n

1<\/p>\n

=<\/p>\n

K<\/p>\n

i<\/p>\n

(<\/p>\n

\u03b1<\/p>\n

i<\/p>\n

)<\/p>\n

,<\/p>\n

\u03b1<\/p>\n

i<\/p>\n

\u2009<\/p>\n

m<\/p>\n

i<\/p>\n

\u2208<\/p>\n

K<\/p>\n

i<\/p>\n

\\\\boxed{ K_{i+1}=K_i(\\\\alpha_i), \\\\qquad \\\\alpha_i^{\\\\,m_i}\\\\in K_i } <\/p>\n

<\/span><\/span><\/span>K<\/span><\/span>i<\/span>+<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>K<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03b1<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>,<\/span><\/span><\/span>\u03b1<\/span><\/span>i<\/span><\/span><\/span><\/span><\/span>m<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2208<\/span><\/span>K<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4e00\u822c\u4e94\u6b21\u65b9\u7a0b\u7684\u4f3d\u7f57\u74e6\u7fa4\u4e3a：<\/p>\n

<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

)<\/p>\n

\u2245<\/p>\n

S<\/p>\n

5<\/p>\n

\\\\boxed{ \\\\operatorname{Gal}(L\/K)\\\\cong S_5 } <\/p>\n

<\/span><\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>K<\/span>)<\/span><\/span>\u2245<\/span><\/span>S<\/span><\/span>5<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d：<\/p>\n

<\/p>\n

\u2223<\/p>\n

S<\/p>\n

5<\/p>\n

\u2223<\/p>\n

=<\/p>\n

5<\/p>\n

!<\/p>\n

=<\/p>\n

120<\/p>\n

|S_5|=5!=120 <\/p>\n

<\/span><\/span>\u2223<\/span>S<\/span><\/span>5<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span><\/span>=<\/span><\/span><\/span><\/span>5<\/span>!<\/span><\/span>=<\/span><\/span><\/span><\/span>120<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u800c：<\/p>\n

<\/p>\n

A<\/p>\n

5<\/p>\n

\u22b4<\/p>\n

S<\/p>\n

5<\/p>\n

\\\\boxed{ A_5\\\\trianglelefteq S_5 } <\/p>\n

<\/span><\/span><\/span>A<\/span><\/span>5<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22b4<\/span><\/span>S<\/span><\/span>5<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5e76\u4e14：<\/p>\n

<\/p>\n

S<\/p>\n

5<\/p>\n

\/<\/p>\n

A<\/p>\n

5<\/p>\n

\u2245<\/p>\n

C<\/p>\n

2<\/p>\n

\\\\boxed{ S_5\/A_5\\\\cong C_2 } <\/p>\n

<\/span><\/span><\/span>S<\/span><\/span>5<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\/<\/span>A<\/span><\/span>5<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2245<\/span><\/span>C<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4f46\u662f：<\/p>\n

<\/p>\n

A<\/p>\n

5<\/p>\n

\u00a0\u4e0d\u662f\u963f\u8d1d\u5c14\u7fa4<\/p>\n

\\\\boxed{ A_5\\\\text{ \u4e0d\u662f\u963f\u8d1d\u5c14\u7fa4} } <\/p>\n

<\/span><\/span><\/span>A<\/span><\/span>5<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00a0<\/span>\u4e0d\u662f\u963f\u8d1d\u5c14\u7fa4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u6b64：<\/p>\n

<\/p>\n

S<\/p>\n

5<\/p>\n

\u00a0\u4e0d\u662f\u53ef\u89e3\u7fa4<\/p>\n

\\\\boxed{ S_5\\\\text{ \u4e0d\u662f\u53ef\u89e3\u7fa4} } <\/p>\n

<\/span><\/span><\/span>S<\/span><\/span>5<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00a0<\/span>\u4e0d\u662f\u53ef\u89e3\u7fa4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\n

\u4e0e\u5224\u522b\u5f0f <\/p>\n

\u0394<\/p>\n

\\\\Delta<\/p>\n

<\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/span> \u7684\u5173\u7cfb<\/h3>\n

\u4e94\u6b21\u65b9\u7a0b\u7684\u5224\u522b\u5f0f\u4e3a：<\/p>\n

<\/p>\n

\u0394<\/p>\n

=<\/p>\n

\u220f<\/p>\n

i<\/p>\n

<<\/p>\n

j<\/p>\n

(<\/p>\n

x<\/p>\n

i<\/p>\n

\u2212<\/p>\n

x<\/p>\n

j<\/p>\n

)<\/p>\n

2<\/p>\n

\\\\boxed{ \\\\Delta= \\\\prod_{i<j}(x_i-x_j)^2 } <\/p>\n

<\/span><\/span><\/span>\u0394<\/span><\/span>=<\/span><\/span><\/span>i<\/span><<\/span>j<\/span><\/span><\/span><\/span><\/span>\u220f<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>x<\/span><\/span>j<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u6b64：<\/p>\n

<\/p>\n

\u0394<\/p>\n

=<\/p>\n

\u00b1<\/p>\n

\u220f<\/p>\n

i<\/p>\n

<<\/p>\n

j<\/p>\n

(<\/p>\n

x<\/p>\n

i<\/p>\n

\u2212<\/p>\n

x<\/p>\n

j<\/p>\n

)<\/p>\n

\\\\boxed{ \\\\sqrt{\\\\Delta} =\\\\pm \\\\prod_{i<j}(x_i-x_j) } <\/p>\n

<\/span><\/span><\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>\u00b1<\/span><\/span><\/span>i<\/span><<\/span>j<\/span><\/span><\/span><\/span><\/span>\u220f<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>x<\/span><\/span>j<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u52a0\u5165 <\/p>\n

\u0394<\/p>\n

\\\\sqrt{\\\\Delta}<\/p>\n

<\/span><\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u540e：<\/p>\n

<\/p>\n

K<\/p>\n

\u2286<\/p>\n

K<\/p>\n

(<\/p>\n

\u0394<\/p>\n

)<\/p>\n

\u2286<\/p>\n

L<\/p>\n

\\\\boxed{ K \\\\subseteq K(\\\\sqrt{\\\\Delta}) \\\\subseteq L } <\/p>\n

<\/span><\/span><\/span>K<\/span><\/span>\u2286<\/span><\/span>K<\/span>(<\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>\u2286<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5bf9\u5e94\u7684\u4f3d\u7f57\u74e6\u7fa4\u4e3a：<\/p>\n

<\/p>\n

S<\/p>\n

5<\/p>\n

\u2287<\/p>\n

A<\/p>\n

5<\/p>\n

\u2287<\/p>\n

{<\/p>\n

e<\/p>\n

}<\/p>\n

\\\\boxed{ S_5 \\\\supseteq A_5 \\\\supseteq \\\\{e\\\\} } <\/p>\n

<\/span><\/span><\/span>S<\/span><\/span>5<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2287<\/span><\/span>A<\/span><\/span>5<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2287<\/span><\/span>{<\/span>e<\/span>}<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u4e3a：<\/p>\n

<\/p>\n

Gal<\/p>\n

\u2061<\/p>\n

(<\/p>\n

L<\/p>\n

\/<\/p>\n

K<\/p>\n

(<\/p>\n

\u0394<\/p>\n

)<\/p>\n

)<\/p>\n

\u2245<\/p>\n

A<\/p>\n

5<\/p>\n

\\\\boxed{ \\\\operatorname{Gal}(L\/K(\\\\sqrt{\\\\Delta})) \\\\cong A_5 } <\/p>\n

<\/span><\/span><\/span>Gal<\/span><\/span>(<\/span>L<\/span>\/<\/span>K<\/span>(<\/span><\/span>\u0394<\/span><\/span><\/span><\/span><\/p>\n

<\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>))<\/span><\/span>\u2245<\/span><\/span>A<\/span><\/span>5<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4f46\u662f <\/p>\n

A<\/p>\n

5<\/p>\n

A_5<\/p>\n

<\/span><\/span>A<\/span><\/span>5<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u662f\u975e\u963f\u8d1d\u5c14\u5355\u7fa4，\u65e0\u6cd5\u7ee7\u7eed\u50cf\u4e09\u6b21\u65b9\u7a0b\u4e2d\u7684 <\/p>\n

A<\/p>\n

3<\/p>\n

A_3<\/p>\n

<\/span><\/span>A<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u90a3\u6837\u901a\u8fc7\u6b63\u89c4\u5b50\u7fa4\u5206\u89e3\u6210\u963f\u8d1d\u5c14\u5546\u7fa4\u3002<\/p>\n

\u6240\u4ee5：<\/p>\n

<\/p>\n

S<\/p>\n

5<\/p>\n

\u00a0\u4e0d\u53ef\u89e3<\/p>\n

\\\\boxed{ S_5\\\\text{ \u4e0d\u53ef\u89e3} } <\/p>\n

<\/span><\/span><\/span>S<\/span><\/span>5<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00a0<\/span>\u4e0d\u53ef\u89e3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

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\n\u57df\u4e0a\u7684\u81ea\u540c\u6784
\n\u8bbe LLL \u662f\u4e00\u4e2a\u57df\u3002
\n\u5982\u679c\u6620\u5c04： \u03c3:L\u27f6L\\\\sigma:L\\\\longrightarrow L \u03c3:L\u27f6L
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