3 Outline

The proof of the criterion has four parts, one per chapter.

Chapter 2 proves that extension of ideals induces an injective map \(\mathrm{Cl}(\mathcal O_{K^+}) \to \mathrm{Cl}(\mathcal O_K)\) (theorem 3.2). Consequently \(h^+ \mid h\), and the relative class number \(h^- = h/h^+\) is a genuine natural number with \(h = h^+ h^-\).

Chapter 3 establishes the analytic formula

\[ h^- = 2p \prod _{\chi \text{ odd}} \Bigl(-\tfrac 12 B_{1,\chi ^{-1}}\Bigr) \]

(theorem 2.5), then reduces it modulo \(p\): indexing the odd characters by powers of the Teichmüller character and applying the congruence \(B_{1,\omega ^j} \equiv B_{j+1}/(j+1) \pmod p\) converts the product into a product of classical divided Bernoulli numbers, giving

\[ p \mid h^- \iff \exists k,\ 1 \le k,\ 2k \le p-3,\ p \mid \operatorname {num}(B_{2k}) \]

(theorem 3.4).

Chapter 4 treats the plus side. The real cyclotomic units form a subgroup \(C^+\) of \((\mathcal O_{K^+})^\times \), and the determinant of the Kummer logarithm matrix — a matrix of \(p\)-adic logarithms of the cyclotomic-unit generators — factors as a product of Bernoulli numbers times a nonzero Vandermonde determinant (theorem 3.7). If no Bernoulli numerator in Kummer’s range is divisible by \(p\), the determinant is nonzero and \(C^+\) is \(p\)-saturated in the full unit group, whence \(p \nmid [(\mathcal O_{K^+})^\times : C^+]\) (theorem 3.12 and theorem 2.7). Sinnott’s prime-conductor index theorem [ 5 ] identifies the \(p\)-divisibility of this index with that of \(h^+\) (theorem 4.5).

Chapter 5 assembles the criterion. The plus-side results yield the implication \(p \mid h^+ \Rightarrow p \mid h^-\) (theorem 1.3); together with \(h = h^+h^-\) this reduces \(p \mid h\) to \(p \mid h^-\), and the minus criterion finishes the proof (theorem 3.1).

Chapter 6 contains the application to the infinitude of irregular primes, following Carlitz [ 1 ] : a finite set \(S\) of primes is escaped by manufacturing an even index \(M\), divisible by \(q - 1\) for every prime \(q \in S\), with \(|B_M/M| {\gt} 1\); any prime in the numerator of \(B_M/M\) is then irregular and lies outside \(S\). The key arithmetic input is the unrestricted Kummer congruence \(B_m/m \equiv B_n/n \pmod p\) for even \(m \equiv n \not\equiv 0 \pmod{p-1}\), proved by the elementary Voronoi route (theorem 5.3).