Herbrand–Ribet theorem

In mathematics, the Herbrand–Ribet theorem is a result on the class group of certain number fields. It is a strengthening of Ernst Kummer's theorem to the effect that the prime p divides the class number of the cyclotomic field of p-th roots of unity if and only if p divides the numerator of the n-th Bernoulli number B_n for some n, 0 < n < p − 1. The Herbrand–Ribet theorem specifies what, in particular, it means when p divides such an B_n.

The Galois group Δ of the cyclotomic field of pth roots of unity for an odd prime p, Q(ζ) with ζ^p = 1, consists of the p − 1 group elements σ_a, where ${\displaystyle \sigma _{a}(\zeta )=\zeta ^{a}}$. As a consequence of Fermat's little theorem, in the ring of p-adic integers ${\displaystyle {\mathbb {Z}}_{p}}$ we have p − 1 roots of unity, each of which is congruent mod p to some number in the range 1 to p − 1; we can therefore define a Dirichlet character ω (the Teichmüller character) with values in ${\displaystyle {\mathbb {Z}}_{p}}$ by requiring that for n relatively prime to p, ω(n) be congruent to n modulo p. The p part of the class group is a ${\displaystyle {\mathbb {Z}}_{p}}$-module (since it is p-primary), hence a module over the group ring ${\displaystyle {\mathbb {Z}}_{p}[\Delta ]}$. We now define idempotent elements of the group ring for each n from 1 to p − 1, as

...
Wikipedia