Let be a cyclotomic extension of , with an odd prime, let be the Sylow -subgroup of the ideal class group of , and let be the Galois group of this extension. Note that the character group of , denoted , is given by
For each , let denote the corresponding orthogonal idempotent of the group ring, and note that the -Sylow subgroup of the ideal class group is a -module under the typical multiplication. Thus, using the orthogonal idempotents, we can decompose the module via .
Last, let denote the th Bernoulli number.
Theorem 1 (Herbrand).
Let be odd with . Then .
Only the first direction of this theorem () was proved by Herbrand himself. The converse is much more intricate, and was proved by Ken Ribet.
|Date of creation||2013-03-22 14:12:45|
|Last modified on||2013-03-22 14:12:45|
|Last modified by||mathcam (2727)|