Herbrand’s theorem

Let (ζp) be a cyclotomic extension of , with p an odd prime, let A be the Sylow p-subgroupMathworldPlanetmathPlanetmath of the ideal class groupPlanetmathPlanetmathPlanetmath of (ζp), and let G be the Galois group of this extension. Note that the character group of G, denoted G^, is given by


For each χG^, let εχ denote the corresponding orthogonal idempotent of the group ringMathworldPlanetmath, and note that the p-Sylow subgroup of the ideal class group is a [G]-module under the typical multiplication. Thus, using the orthogonal idempotents, we can decompose the module A via A=i=0p-2Aωii=0p-2Ai.

Last, let Bk denote the kth Bernoulli numberMathworldPlanetmathPlanetmath.

Theorem 1 (Herbrand).

Let i be odd with 3ip-2. Then Ai0pBp-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.

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