Frobenius element

Summary

Let $L/K$ be a Galois extension of number fields, and let $\mathfrak{P}$ be a prime of $L$ that is unramified over $K$. Let $q$ be the size of the residue field of the prime of $K$ below $\mathfrak{P}$. The Frobenius element of $\mathfrak{P}$ is the unique element of the decomposition group that acts on the residue field as the Frobenius automorphism $x \mapsto x^q$. Its conjugacy class depends only on the prime of $K$ below $\mathfrak{P}$. Frobenius elements encode how primes split, and they are the input of the Artin map in class field theory and of Chebotarev's density theorem.