# 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 $\mathcal{O}_{F}^{\times}$ denote the group of units in the ring of integers of $F$, let $C$ 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, $N$ 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 $[F:\mathbb{Q}]$ and suppose $\theta\in\mathbb{Z}[G]$ annihilates the Sylow $p$-subgroup of $E/C^{\prime}$. Then $2\theta$ 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).

Title Thaine’s theorem ThainesTheorem 2013-03-22 14:12:34 2013-03-22 14:12:34 mathcam (2727) mathcam (2727) 6 mathcam (2727) Theorem msc 11R29