2 The relative class-number formula

Theorem 2.1

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The class number \(h^+\) is expressed by the even part of the cyclotomic \(L\)-product with the explicit cyclotomic \(K^+\) factor:

\[ h^+(K)= \operatorname {factor}_{K^+} \prod _{\chi \ \mathrm{even},\ \chi \ne 1} L(1,\chi ). \]

Proof ▶

The proof rewrites the analytic class-number formula for the maximal real subfield using the product over even non-trivial characters.

Theorem 2.2

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The product of the Gauss sums over odd characters has the explicit value used to cancel the analytic constants in the relative class-number formula.

Proof ▶

The proof splits into the cases \(p\equiv 1\pmod4\) and \(p\equiv 3\pmod4\). Non-quadratic odd characters are paired with their inverses, using the usual identity for a Gauss sum times its inverse. The possible quadratic odd character contributes the classical quadratic Gauss sum.

Theorem 2.3

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The explicit Gauss-product formula satisfies the packaged Gauss hypothesis needed by the relative class-number assembly theorem.

Proof ▶

The raw product identity is rewritten into the Gauss-product hypothesis needed by the relative class-number formula.

Theorem 2.4

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Assume the cyclotomic residue factorisation, the \(h^+\) even-value formula, and the packaged Gauss-product identity. Then

\[ h^-(K) = 2p\prod _{\chi \ \mathrm{odd}} \left(-\frac12 B_{1,\chi ^{-1}}\right). \]

Proof ▶

The proof divides the full cyclotomic class-number formula by the \(K^+\) formula, uses the residue factorisation to leave only the odd \(L\)-values, and then substitutes the odd \(L(1,\chi )\) formula. The Gauss product cancels the remaining analytic constants, leaving exactly the product of generalized Bernoulli factors.

Theorem 2.5

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For an odd prime cyclotomic field,

\[ h^-(K) = 2p\prod _{\chi \ \mathrm{odd}} \left(-\frac12 B_{1,\chi ^{-1}}\right). \]

Proof ▶

The proof supplies the three inputs of theorem 2.4: the Dedekind-zeta residue factorisation into even and odd \(L\)-products, theorem 2.1, and theorem 2.3.