We need some notation. We write *K <= M* to mean that *K*
and *M* are fields and that *M* is a field extension of *K*.
Associated to such an extension we have the Galois group Gal(*M/K)*).
This raises the question if we have an intermediate field
*K <= L <= M* what can we say about Gal(*M/L*) and
Gal(*L/K*)? Galois theory gives us a good answer to this question.

First notice that Gal(*M/L*) is a subgroup of Gal(*M/K*)
which we shall write as Gal(*M/L*) <= Gal(*M/K*), for short.

**Proof:** To see this recall that, by definition, an element of Gal(*M/L*)
is an *L*-automorphism of *M*. Since *L* contains
*K* any such automorphism must fix the elements of *K*.
Thus we can see that Gal(*M/L*) is a subset of Gal(*M/K*).
But it is clear that if we compose two *L*-automorphisms
or invert an *L*-automorphism then we obtain another one.
Thus it is a subgroup, as required.

This result gives us a way to turn a picture involving fields into
a picture involving groups. We just apply Gal(*M/-*).
But does this process retain information or destroy it? It turns out
that to retain information we must insist that the field extension
*K <= M* has some particular properties.

**Defintion**
We say that a field extension *K <= M*
is **Galois** if it is finite-dimensional, normal
and separable.

As an example, if *M* is a splitting field over *K*
of a separable polynomial then it is Galois.

We need some notation. Write
*[M:K]* for the dimension of *M* considered
as a *K*-vector space. If *G* is a subgroup
of Gal(*M/K*) we write *M ^{G}* for the subset

*{m*in

*M : g(m)=m*for all

*g*in

*G}*. This is called the

**fixed field**of

*G*and, as its name suggests, we have

*K <= M*.

^{G}<= MHere's the first main theorem.

**Theorem** Let *K <= M* be a Galois field extension.
Then

- |Gal (
*M/K*)|=*[L:K]* -
There is an bijection between fields
*L*with*K <= L < M*and subgroups of Gal(*M/K*).

The bijection works like this. If*L*is such a field the corresponding subgroup is Gal(*M/L*). If*G*is such a group the corresponding field is*M*^{G}

Note that these bijections turn a *<=* into a *>=*
because as the field *L* gets bigger Gal(*M/L*) gets
smaller and as the group *G* gets bigger the field
*M ^{G}* gets smaller.

Let's consider an example. Let *a* be the real cube root
of 2 and let *w=e ^{2pii/3}* be a primitive complex
root of unity. The field extension

*K*=

**Q**<=

**Q**(

*a,w*)=

*L*is Galois (

*L*is the splitting field of

*x*over

^{3}-2**Q**) so we can relate the subfields of

*L*to the subgroups of Gal(

*L/K*). Firstly then we should compute the Galois group. Let

*f*be an element of Gal(

*L/K*). Then

*f(a)*must be a root of the minimal polynomial of

*a*over

**Q**, which is

*x*. Thus

^{3}-2*f(a)*is one of

*a,wa,w*. Likewise

^{2}a*f(w)*must be one of

*w*and

*w*. Since a

^{2}**Q**-automorphism is determined by its value on

*a*and

*w*then this means that there are at most six elements in the group. By the theorem (since [L:K]=6) we know that there are 6 automorphisms in the group. So these 6 possibilities all occur. Let

*g*be the

**Q**-automorphism with

*g(a)=aw*and

*g(w)=w*and let

*h*be the

**Q**-automorphism with

*h(a)=a*and

*h(w)=w*. Then the six elements of the Galois group are

^{2}*1,g,g*. Note that the group is not abelian because

^{2},h,gh,g^{2}h*gh(a)=g(h(a))=g(a)=aw*and

*hg(a)=h(g(a))=h(aw)=h(a)h(w)=aw*. Thus

^{2}*gh*is not equal to

*hg*. But there is only one group of order 6 that is not abelian the symmetric group

*S*of permutations of three objects. Can we see three natural things that the Galois group permutes? The answer is yes:

_{3}*a,aw,aw*.

^{2}
What subfields does **Q**(*a,w*) have? Well it's not too hard to
compute them, they are **Q**(*a*), **Q**(*aw*),
**Q**(*aw ^{2}*),

**Q**(

*w*),

**Q**(

*w*) and

^{2}**Q**. See if you can figure out the corresponding subgroups of the Galois group. For example, the first subfield I listed corresponds to the subgroup

*<h>*.

The next main theorem tells us about normal subgroups of the Galois group which are the most interesting kind of subgroups.

**Theorem** Let *K < L <= M* with *M* a Galois
extension of *K*.

TFAE

*K <= L*is Galois*[L:K]*=|Gal(*L/K*)|- |Gal(
*M/L*)| is a normal subgroup of |Gal(*M/K*)|

*M/K*)/Gal(

*M/L*) is isomorphic to Gal(

*L/K*)

These are not the only theorems in Galois theory but they are the most basic ones.