# 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 ($\u27f9$) was proved by Herbrand himself. The converse is much more intricate, and was proved by Ken Ribet.

Title | Herbrand’s theorem |
---|---|

Canonical name | HerbrandsTheorem |

Date of creation | 2013-03-22 14:12:45 |

Last modified on | 2013-03-22 14:12:45 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 5 |

Author | mathcam (2727) |

Entry type | Theorem |

Classification | msc 11R29 |