If
is a number field, then the *Galois closure*
of in
is the field generated by all images of under all
embeddings in
(more generally, if is an extension, the
Galois closure of over is the field generated by images of
embeddings
that are the identity map on ). If
,
then is the field generated by all of the conjugates of , and is
hence Galois over
, since the image under an embedding of any
polynomial in the conjugates of is again a polynomial in
conjugates of .

How much bigger can the degree of be as compared to the degree of ? There is an embedding of into the group of permutations of the conjugates of . If has conjugates, then this is an embedding , where is the symmetric group on symbols, which has order . Thus the degree of the over is a divisor of . Also is a transitive subgroup of , which constrains the possibilities further. When , we recover the fact that quadratic extensions are Galois. When , we see that the Galois closure of a cubic extension is either the cubic extension or a quadratic extension of the cubic extension. One can show that the Galois closure of a cubic extension is obtained by adjoining the square root of the discriminant, which is why an irreducible cubic defines a Galois extension if and only if the discriminant is a perfect square.

For an extension of of degree , it is ``frequently'' the case that the Galois closure has degree , and in fact it is an interesting problem to enumerate examples of degree extension in which the Galois closure has degree smaller than . For example, the only possibilities for the order of a transitive proper subgroup of are , , , and ; there are also proper subgroups of order , and , but none are transitive.

Let be a positive integer. Consider the field
,
where
is a primitive th root of unity. If
is an embedding, then
is also an
th root of unity, and the group of th roots of unity is cyclic,
so
for some which is invertible
modulo . Thus is Galois and
.
However,
, so this map is an isomorphism. (Remark:
Taking a limit using the maps
, we obtain a homomorphism
, which is called the *-adic cyclotomic character*.)

Compositums of Galois extensions are Galois. For example, the biquadratic field is a Galois extension of of degree , which is the compositum of the Galois extensions and of .

Fix a number field that is Galois over a subfield . Then the Galois group acts on many of the object that we have associated to , including:

- the integers ,
- the units ,
- the group of fractional ideals of ,
- the class group , and
- the set of prime ideals lying over a given nonzero prime ideal of , i.e., the prime divisors of .

William Stein 2012-09-24