The Analytic Class Number Formula

The class number is a useful invariant for a number field which gives a sense of how much unique factorization fails in that number field's ring of integers. We will be proving the analytic class number formula, which is a formula relating many invariants of a number field (including the class number) to the residue of its Dedekind zeta function at 1. This formula, used to show Dedekind zeta functions have simple poles at 1, is also used to prove Dirichlet's theorem on primes in arithmetic progressions.