# 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 $\hat{G}$, is given by

 $\displaystyle\hat{G}=\{\chi^{i}\mid 0\leq i\leq p-2\}$

For each $\chi\in\hat{G}$, let $\varepsilon_{\chi}$ 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 $3\leq i\leq p-2$. Then $A_{i}\neq 0\iff p\mid B_{p-i}$.

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

Title Herbrand’s theorem HerbrandsTheorem 2013-03-22 14:12:45 2013-03-22 14:12:45 mathcam (2727) mathcam (2727) 5 mathcam (2727) Theorem msc 11R29