The fundamental theorem of Galois theory states that for a finite Galois extension field 
of a field 
, 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
of 
iff 
is a Galois extension
field. Since any subfield of a separable
extension, which the Galois extension field 
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 
.
For example, let 
and 
, with intermediate subfields 
,
and 
.
Their corresponding subgroups are 
, 
and 
. The only field
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, let 
be a primitive root
of unity of order 3 and consider the Galois
extension field
 |  |  |
(4)
|
 |  |  |
(5)
|
over 
,
which has extension field degree six. That
is, it is a six-dimensional vector
space over the rationals.
