Galois theory applied to prime decomposition

Abstract

Up to now the Galois-theoretic aspects of number fields have not figured prominently in our theory. Essentially all we did was to determine the Galois group of the mth cyclotomic field (it was the multiplicative group of integers mod m) and to show that, in the case of a normal extension, the Galois group permutes the primes over a given prime transitively (Theorem 23). Galois groups also turned up in the proof of Theorem 26 on splitting in cyclotomic fields. In this chapter we apply Galois theory to the general problem of determining how a prime ideal of a number ring splits in an extension field.