Splitting of prime ideals

Summary

Let $L/K$ be an extension of algebraic number fields. A prime ideal of the ring of integers of $K$ need not stay prime in the ring of integers of $L$. It factors into a product of prime ideals of $L$. The factorisation is described by the number of factors, their exponents, called ramification indices, and their residue degrees. These numbers satisfy the fundamental identity $\sum_i e_i f_i = [L : K]$. In a Galois extension the splitting is controlled by the decomposition group and the Frobenius element. This leads to Chebotarev's density theorem and to class field theory.