Thaine’s theorem

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

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

for some collection of $b_a\in \mathbb{Z}$. (Here, $N$ denotes the norm operator and ${\zeta }_m$ is a primitive $m$-th root of unity.) Finally, let $A$ denote the ideal class group of $F$.

Theorem 1 (Thaine).

Suppose $p$ is a rational prime not dividing the degree $\mathrm{[}F\mathrm{:}\mathrm{Q}\mathrm{]}$ and suppose $\theta \mathrm{\in }\mathrm{Z}\mathit{}\mathrm{[}G\mathrm{]}$ annihilates the Sylow $p$-subgroup of $E\mathrm{/}{C}^{\mathrm{\prime }}$. Then $\mathrm{2}\mathit{}\theta$ annihilates the Sylow $p$-subgroup of $A$.

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).