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Herbrand's theorem
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(Theorem)
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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
![$ \mathbb{Z}[G]$](http://images.planetmath.org:8080/cache/objects/5647/l2h/img14.png "$ \mathbb{Z}[G]$") -module under the typical multiplication. Thus, using the orthogonal idempotents, we can decompose the module  via
 .
Last, let  denote the  th Bernoulli number.
Only the first direction of this theorem ( ) was proved by Herbrand himself. The converse is much more intricate, and was proved by Ken Ribet.
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"Herbrand's theorem" is owned by mathcam.
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Cross-references: converse, Bernoulli number, module, orthogonal idempotents, multiplication, subgroup, orthogonal idempotent of the group ring, group, character, extension, Galois group, ideal class group, prime, odd, cyclotomic extension
There are 3 references to this entry.
This is version 2 of Herbrand's theorem, born on 2004-02-27, modified 2004-03-05.
Object id is 5647, canonical name is HerbrandsTheorem.
Accessed 2230 times total.
Classification:
| AMS MSC: | 11R29 (Number theory :: Algebraic number theory: global fields :: Class numbers, class groups, discriminants) |
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Pending Errata and Addenda
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