# Galois Extensions

In this section we give a survey (no proofs) of the basic facts about Galois extensions of that will be needed in the rest of this chapter.

Definition 9.1.1 (Galois)   An extension of number fields is Galois if

where is the group of automorphisms of that fix . We write

For example, if is a number field embedded in the complex numbers, then is Galois over if every field homomorphism has image . As another example, any quadratic extension is Galois over , since it is of the form , for some , and the nontrivial automorphism is induced by , so there is always one nontrivial automorphism. If is an irreducible cubic polynomial, and is a root of , then one proves in a course on Galois theory that is Galois over if and only if the discriminant of  is a perfect square in . Random'' number fields of degree bigger than are rarely Galois.

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 .
In the next section we will be concerned with the action of on , though actions on each of the other objects, especially , are also of great interest. Understanding the action of on will enable us to associate, in a natural way, a holomorphic -function to any complex representation .

William Stein 2012-09-24