# Galois Extensions

Suppose is a number field. Then is if every field homomorphism has image , or equivalently, . More generally, we have the following definition.

Definition 13.1.1 (Galois)   An extension of number fields is if , where is the group of automorphisms of that fix . We write .

For example, is Galois (over itself), any quadratic extension is Galois, since it is of the form , for some , and the nontrivial embedding 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 in 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 (I will not justify this claim further in this course).

If is a number field, then the Galois closure of 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 generated by each 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 a natural embedding of into the group of permutations of the conjugates of . If there are conjugates of , 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 the Galois group 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. It turns out that that Galois closure of a cubic extension is obtained by adjoining the square root of the discriminant. For an extension of degree , it is frequently'' the case that the Galois closure has degree , and in fact it is a difficult and interesting problem to find examples of degree extension in which the Galois closure has degree smaller than (according to : the only possibilities for the order of a transitive proper subgroup of are , , , and ; there are five transitive subgroups of out of the total of subgroups of ).

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. (Side note: Taking a -adic limit and 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 .

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 nonzero fractional ideals of ,
• the class group , and
• the set of prime ideals lying over a given prime of .
In the next section we will be concerned with the action of on , though actions on each of the other objects, especially , will be of further interest.

William Stein 2004-05-06