Stickelberger's theorem
StickelbergersTheorem
2004-02-27 12:46:57
2004-03-02 17:01:56
Theorem
Stickelberger element
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\begin{Theo}[Stickelberger]
Let $L=\mathbb{Q}(\zeta_m)$ be a cyclotomic field extension of $\Q$ with Galois group $G=\{\sigma_a\}_{a\in(\Z/m\Z)^\times}$, and consider the group ring $\Q[G]$. Define the Stickelberger element $\theta\in\Q[G]$ by
\begin{align*}
\theta=\frac{1}{m}\sum_{1\leq a\leq m, (a,m)=1}a\sigma_a^{-1},
\end{align*}
and take $\beta\in\Z[G]$ such that $\beta\theta\in\Z[G]$ as well. Then $\beta\theta$ is an annihilator for the ideal class group of $\Q(\zeta_m)$.
\end{Theo}
Note that $\theta$ itself need not be an annihilator, just that any multiple of it in $\Z[G]$ is.
This result allows for the most basic \PMlinkescapetext{connections} between the (otherwise hard to determine) \PMlinkescapetext{structure} of a cyclotomic ideal class group and its \PMlinkescapetext{collection} of annihilators. For an application of Stickelberger's theorem, see Herbrand's theorem.