class number formula for quadratic fields in 1839, but it was stated in the language of quadratic forms rather than classes of ideals. It appears that Gauss already knew this formula in 1801. This exposition follows Davenport. Let d be a fundamental discriminant, and write h(d) for the number of equivalence classes of quadratic forms with discriminant d. Let be the Kronecker symbol. Then is a Dirichlet character. Write for the Dirichlet L-series based on . For d > 0, let t > 0, u > 0 be the solution to the Pell equation for which u is smallest, and write (Then is either a fundamental unit of the real quadratic field or the square of a fundamental unit.) For d < 0, write w for the number of automorphisms of quadratic forms of discriminant d; that is, Then Dirichlet showed that This is a special case of Theorem 1 above: for a quadratic field K, the Dedekind zeta function is just , and the residue is . Dirichlet also showed that the L-series can be written in a finite form, which gives a finite form for the class number. Suppose is primitive with prime conductor . Then Galois extensions of the rationals If K is a Galois extension of Q, the theory of Artin L-functions applies to . It has one factor of the Riemann zeta function, which has a pole of residue one, and the quotient is regular at s = 1. This means that the right-hand side of the class number formula can be equated to a left-hand side Π L(1,ρ)dim ρ with ρ running over the classes of irreducible non-trivial complex linear representations of Gal(K/Q) of dimension dim(ρ). That is according to the standard decomposition of the regular representation. Abelian extensions of the rationals This is the case of the above, with Gal(K/Q) an abelian group, in which all the ρ can be replaced by Dirichlet characters (via class field theory) for some modulus f called the conductor. Therefore all the L(1) values occur for Dirichlet L-functions, for which there is a classical formula, involving logarithms. By the Kronecker–Weber theorem, all the values required for an analytic class number formula occur already when the cyclotomic fields are considered. In that case there is a further formulation possible, as shown by Kummer. The regulator, a calculation of volume in 'logarithmic space' as divided by the logarithms of