TITLE: History of the analytic class number formula
QUESTION [17 upvotes]: The (general) analytic class number formula gives a value for the residue of the Dedekind zeta function of a number field at the point $s=1$ (or, as I prefer, the leading Taylor coefficient at $s=0$). To whom should this formula be attributed?
My usual go-to place for such history questions is Narkiewicz's book Elementary and analytic theory of algebraic numbers, but it only discusses the abelian case. It lists references as late as 1929 for the result in the abelian case. Does one have to wait for Artin or Hasse?

REPLY [10 votes]: KConrad's answer is correct, and the analytic class number formula is due to Dedekind. Yet the whole story is a little bit more complex and it is fair to say
that Dedekind's analytic class number formula is essentially due to Dirichlet and Kummer, in the following sense. Dirichlet (who had already proved the class number formula for quadratic forms in his proof on the theorem of primes in arithmetic progression) determined the number of equivalence classes of forms that split into linear factors over ${\mathbb Q}(\zeta_p)$ (a key idea of which he had during a mass in the Sistine chapel), but did not publish his results as he knew that Kummer was working out a similar formula using his new language of ideal numbers. The only reason why Kummer did not (and could not) prove the general class number formula was that his theory of ideal numbers did not seem to work in general number rings. What Kummer was missing was the definition of an algebraic integer, and at this point Dedekind enters the story. He worked out a theory of ideals in algebraic rings of integers; transferring Kummer's analytic class number formula to the general case did not require any really new idea. In no way should these comments belittle Dedekind's contributions to algebraic number theory.