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Herbrand's theorem
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(Theorem)
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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
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.
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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 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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