PlanetMath: Thaine's theorem

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Thaine's theorem

(Theorem)

Let $F/\mathbb{Q}$ be a totally real abelian number field. By the Kronecker-Weber theorem, there exists an such that $F\subset \mathbb{Q}(\zeta_m)$ . Let be the Galois group of the extension . Let $\mathcal {O}_F^\times$ denote the group of units in the ring of integers of , let be the subgroup of $\mathcal {O}_F^\times$ consisting of units $\eta$ of the form

$\displaystyle \eta=\pm N_{\mathbb{Q}(\zeta_m)/F}\left(\prod_{a\in(\mathbb{Z}/m\mathbb{Z})^\times}(\zeta_m^a-1)^{b_a}\right)$

for some collection of $b_a\in\mathbb{Z}$ . (Here,

denotes the norm operator and $\zeta_m$ is a primitive

-th root of unity.) Finally, let

denote the ideal class group of

Theorem 1 (Thaine) Suppose is a rational prime not dividing the degree $[F:\mathbb{Q}]$ and suppose $\theta\in\mathbb{Z}[G]$ annihilates the Sylow -subgroup of . Then $2\theta$ annihilates the Sylow -subgroup of .

This is one of the most sophisticated results concerning the annihilators of an ideal class group. It is a direct, but more complicated, version of Stickelberger's theorem, applied to totally real fields (for which Stickelberger's theorem gives no information).

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Cross-references: information, totally real fields, Stickelberger's theorem, annihilators, degree, rational prime, ideal class group, root of unity, primitive, operator, norm, collection, units, subgroup, ring of integers, group of units, extension, Galois group, Kronecker-Weber theorem, abelian number field, real
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This is version 3 of Thaine's theorem, born on 2004-02-27, modified 2004-03-02.
Object id is 5643, canonical name is ThainesTheorem.
Accessed 1644 times total.

Classification:

AMS MSC:

11R29 (Number theory :: Algebraic number theory: global fields :: Class numbers, class groups, discriminants)

Pending Errata and Addenda

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