Thaine's theorem
ThainesTheorem
2004-02-27 15:23:20
2004-03-02 17:12:40
Theorem
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Let $F/\Q$ be a totally real abelian number field. By the Kronecker-Weber theorem, there exists an $m$ such that $F\subset \Q(\zeta_m)$. Let $G$ be the Galois group of the extension $F/Q$. Let $\mc{O}_F^\times$ denote the group of units in the ring of integers of $F$, let $C$ be the subgroup of $\mc{O}_F^\times$ consisting of units $\eta$ of the form
\begin{align*}
\eta=\pm N_{\Q(\zeta_m)/F}\left(\prod_{a\in(\Z/m\Z)^\times}(\zeta_m^a-1)^{b_a}\right)
\end{align*}
for some collection of $b_a\in\Z$. (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 $F$.
\begin{theorem}[Thaine]
Suppose $p$ is a rational prime not dividing the degree $[F:\Q]$ and suppose $\theta\in\Z[G]$ annihilates the Sylow $p$-subgroup of $E/C'$. Then $2\theta$ annihilates the Sylow $p$-subgroup of $A$.
\end{theorem}
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).