A Galois extension field 
of 
is a normal extension
that is also a separable extension. For an
algebraic extension 
, the following definitions are equivalent.
1. 
is the splitting field for a collection of separable polynomials. When 
is a finite extension,
then only one separable polynomial is necessary.
2. The common fixed field of all field automorphisms of 
that fix 
is exactly 
.
3. Every irreducible polynomial over 
which has a root
in 
factors into linear factors in 
. Also, 
must be a separable extension.
4. Every field automorphism 
of the algebraic
closure 
that fixes 
satisfies 
.
Also, 
must be a separable extension.
A Galois extension has all of the above properties. For example, consider 
, the rational numbers
adjoined by the imaginary unit 
, over 
, which is a Galois extension. Note that 
contains all of the roots
of 
,
and is generated by them, so it is the splitting field
of 
.
Of course, there are two distinct roots in 
so it is separable. The only nontrivial
automorphism fixing 
is given by complex conjugation
 |
(1)
|
whose fixed field is 
. The only monic irreducible
polynomials with rational coefficients
with roots of the form 
with 
are 
(
) and 
(
). Both split into linear factors over 
. Finally, the algebraic closure 
is the set
of algebraic numbers in 
. Every field automorphism
of the algebraic numbers fixing 
sends 
to 
or 
and therefore preserves 
. In general, it is not so simple to verify all of these
properties, which makes their equivalence useful.
There are a couple of ways for an extension not to be a Galois extension. One is for it to not be a normal extension. For instance,
is not normal, and hence not Galois. It is missing the complex roots
of 
.
Its only nontrivial automorphism is defined by 
, which not only
fixes 
but also the subfield 
.
Another possibility for a non-Galois extension is for it to be not separable. The field characteristic of such a field
must be positive since all irreducible
polynomials are separable in characteristic zero. Moreover, all finite fields
are perfect, i.e., all algebraic
extensions are separable extensions. Consider
the field of rational
functions with coefficients in 
, infinite in size and
characteristic 2 (
).
 |
(2)
|
and the extension
 |
(3)
|
For instance, 
and 
.
Then 
is the splitting field of 
, as 
in characteristic
2, and so it is a normal extension. However,
is not separable because 
has repeated roots in
its splitting field, 
.
