伽罗瓦理论基本定理[Fundamental Theorem of Galois Theory][2005-11-8] iamet 发表在 ∑〖数学〗
For a Galois extension of a field, the fundamental theorem of Galois theory states that the subgroups of the Galois group correspond with the subfields of containing . If the subfield corresponds to the subgroup , then the extension field degree of over is the group order of ,
(1)
(2)
Suppose , then and correspond to subgroups and of such that is a subgroup of . Also, is a normal subgroup iff is a Galois extension. Since any subfield of a separable extension, which the Galois extension must be, is also separable, is Galois iff is a normal extension of . So normal extensions correspond to normal subgroups. When is normal, then
(3)
as the quotient group of the group action of on . According to the fundamental theorem, there is a one-one correspondence between subgroups of the Galois group and subfields of containing . For example, for the number field shown above, the only automorphisms of (keeping fixed) are the identity, , , and , so these form the Galois group (which is generated by and ). In particular, the generators and of are as follows: maps to , to , and fixes ; maps to , to and fixes ; and maps to , to and fixes . For example, consider the Galois extension
(4)
(5)
over , which has extension field degree six. That is, it is a six-dimensional vector space over the rationals.
of a field 
, the fundamental theorem of Galois theory states that the subgroups of the Galois group 
correspond with the subfields of 
containing 
. If the subfield 
corresponds to the subgroup 
, then the extension field degree of 
over 
is the group order of 
,
, then 
and 
correspond to subgroups 
and 
of 
such that 
is a subgroup of 
. Also,
is a normal subgroup iff 
is a Galois extension. Since any subfield of a separable extension, which the Galois extension 
must be, is also separable,
is Galois iff 
is a normal extension of 
. So normal extensions correspond to normal subgroups. When 
is normal, then 
on 
.
and subfields of 
containing 
. For example, for the number field 
shown above, the only automorphisms of 
(keeping 
fixed) are the identity,
, 
, and 
, so these form the Galois group 
(which is generated by 
and 
). In particular, the generators 
and 
of 
are as follows:
maps 
to 
,
to 
, and fixes 
;
maps 
to 
,
to 
and fixes 
; and 
maps 
to 
,
to 
and fixes 
.
, which has extension field degree six. That is, it is a six-dimensional vector space over the rationals.