2 The relative class-number formula
The plus class number is given by the even part of the cyclotomic \(L\)-value product:
where each factor of the product equals \(L(1,\chi )\) and \(c_{K^+}\) is the explicit constant of the analytic class-number formula of \(K^+\) (built from the discriminant, the regulator and the roots of unity of \(K^+\)).
The analytic class-number formula for the totally real field \(K^+\) expresses \(h^+\) through the residue of the Dedekind zeta function \(\zeta _{K^+}\). Factoring \(\zeta _{K^+}(s) = \zeta (s)\prod _{\chi \text{ even},\ \chi \ne 1} L(s,\chi )\) over the even characters and inserting the classical evaluation of \(L(1,\chi )\) for even primitive \(\chi \) — the displayed Gauss-sum-and-logarithm expression — gives the formula.
The product of the Gauss sums over the odd characters modulo \(p\) is
The proof splits into the cases \(p\equiv 1\pmod4\) and \(p\equiv 3\pmod4\). A non-quadratic odd character \(\chi \) is paired with its distinct inverse \(\chi ^{-1}\) (also odd), and the identity \(\tau (\chi )\tau (\chi ^{-1}) = \chi (-1)\, p = -p\) evaluates each pair. When \(p \equiv 3 \pmod4\) the quadratic character is odd and unpaired; it contributes the classical quadratic Gauss sum \(i\sqrt p\). Counting pairs in the two cases and collecting signs and powers of \(i\) gives the stated value.
The explicit Gauss-product evaluation of theorem 2.2 satisfies the packaged Gauss hypothesis required by the relative class-number assembly theorem (theorem 2.4).
Purely formal: the assembly theorem consumes the Gauss product in a normalised shape (with the analytic constants of the odd \(L\)-value formula distributed over the product), and this normalisation is a finite algebraic rewriting of theorem 2.2.
Assume the factorisation of the Dedekind zeta residue of \(K\) into the even and odd \(L\)-value products, the \(h^+\) formula of theorem 2.1, and the packaged Gauss-product identity. Then
Divide the analytic class-number formula of \(K\) (in residue form) by that of \(K^+\). Using \(h = h^+ h^-\), the quotient expresses \(h^-\) times the ratio of the analytic constants as the product of the odd \(L\)-values \(L(1,\chi )\). Substituting the odd \(L\)-value formula
for each odd \(\chi \) and cancelling the powers of \(\pi \), \(i\) and the Gauss sums against the analytic constants — this is exactly what the packaged Gauss-product identity provides — leaves the displayed product of generalised Bernoulli factors with the prefactor \(2p\).
For an odd prime cyclotomic field,
Supply the three inputs of theorem 2.4: the residue of \(\zeta _K\) factors over the nontrivial characters into the even and odd \(L\)-value products; the even product is converted by theorem 2.1; and the Gauss hypothesis is theorem 2.3.