PlanetMath (more info)
 Math for the people, by the people. Sponsor PlanetMath
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
Owner confidence rating: Very high Entry average rating: No information on entry rating
Herbrand's theorem (Theorem)

Let $\mb{Q}(\zeta_p)$ be a cyclotomic extension of $\Q$ , with $p$ an odd prime, let $A$ be the Sylow $p$ -subgroup of the ideal class group of $\mb{Q}(\zeta_p)$ , and let $G$ be the Galois group of this extension. Note that the character group of $G$ , denoted $\hat{G}$ , is given by

$\displaystyle \hat{G}=\{\chi^i\mid0\leq i\leq p-2\}$    

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.




"Herbrand's theorem" is owned by mathcam.
(view preamble | get metadata)

View style:

Log in to rate this entry.
(view current ratings)

Cross-references: converse, theorem, 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 2751 times total.

Classification:
AMS MSC11R29 (Number theory :: Algebraic number theory: global fields :: Class numbers, class groups, discriminants)

Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy
The character group $\hat{G}$ by micah on 2007-06-15 15:02:12
Forgive me if this suggestion is too pedantic, but I think that it might be useful to specify some $\chi$ that generates the group $\hat{G}$. While I can certainly deduce that such a $\chi$ needs to be some non-trivial complex character, and can then deduce that any such non-trivial character will do, it might be useful to state something to this effect.
[ reply | up ]

Interact
post | correct | update request | prove | add result | add corollary | add example | add (any)