# Decomposition of Primes

Fix a prime and write , so .

Definition 13.2.1 (Residue class degree)   Suppose is a prime of lying over . Then the of is

i.e., the degree of the extension of residue class fields.

If is a tower of field extensions and is a prime of over , then

so the residue class degree is multiplicative in towers.

Note that if and , then induces an isomorphism of finite fields that fixes the common subfield . Thus the residue class degrees of and are the same. In fact, much more is true.

Theorem 13.2.2   Suppose is a Galois extension of number fields, and let be a prime of . Write , and let . Then acts transitively on the set of primes ,

and , where is the common value of the and is the common value of the .

Proof. For simplicity, we will give the proof only in the case , but the proof works in general. Suppose and , and . We will first prove that acts transitively on . Let for some . Recall that we proved long ago, using the Chinese Remainder Theorem (Theorem 9.1.3) that there exists such that is an integral ideal that is coprime to . The product

 (13.1)

is a nonzero integral ideal since it is a product of nonzero integral ideals. Since we have that . Thus the numerator of the rightmost expression in (13.2.1) is divisible by . Also, because is coprime to , each is coprime to as well. Thus is coprime to . Thus the denominator of the rightmost expression in (13.2.1) must also be divisibly by in order to cancel the in the numerator. Thus for any we have

which in particular implies that acts transitively on the .

Choose some and suppose that is another index. Because acts transitively, there exists such that . Applying to the factorization , we see that

Taking on both sides we get . Thus .

As was mentioned right before the statement of the theorem, for any we have , so by transitivity . Since is a lattice in , we have

which completes the proof.

The rest of this section illustrates the theorem for quadratic fields and a cubic field and its Galois closure.

Subsections
William Stein 2004-05-06