Herbrand’s theorem

Let $\mathbb{Q}({\zeta }_p)$ be a cyclotomic extension of $\mathbb{Q}$, with $p$ an odd prime, let $A$ be the Sylow $p$-subgroup of the ideal class group of $\mathbb{Q}({\zeta }_p)$, and let $G$ be the Galois group of this extension. Note that the character group of $G$, denoted $\widehat{G}$, is given by

$\widehat{G}=\{{\chi }^i\mid 0\le i\le p-2\}$

For each $\chi \in \widehat{G}$, let ${\epsilon }_{\chi }$ denote the corresponding orthogonal idempotent of the group ring, and note that the $p$-Sylow subgroup of the ideal class group is a $\mathbb{Z}[G]$-module under the typical multiplication. Thus, using the orthogonal idempotents, we can decompose the module $A$ via $A={\sum }_{i=0}^{p-2}{A}_{{\omega }^i}\equiv {\sum }_{i=0}^{p-2}{A}_i$.

Last, let $B_k$ denote the $k$th Bernoulli number.

Theorem 1 (Herbrand).

Let $i$ be odd with $\mathrm{3}\mathrm{\le }i\mathrm{\le }p\mathrm{-}\mathrm{2}$. Then $A_i\mathrm{\ne }\mathrm{0}\mathrm{\iff }p\mathrm{\mid }{B}_{p\mathrm{-}i}$.

Only the first direction of this theorem ($⟹$) was proved by Herbrand himself. The converse is much more intricate, and was proved by Ken Ribet.