Stickelberger’s theorem

Theorem 1 (Stickelberger).

Let $L\mathrm{=}\mathrm{Q}\mathit{}\mathrm{(}{\zeta }_m\mathrm{)}$ be a cyclotomic field extension of $\mathrm{Q}$ with Galois group $G\mathrm{=}{\mathrm{\{}{\sigma }_a\mathrm{\}}}_{a\mathrm{\in }{\mathrm{(}\mathrm{Z}\mathrm{/}m\mathit{}\mathrm{Z}\mathrm{)}}^{\mathrm{\times }}}$, and consider the group ring $\mathrm{Q}\mathit{}\mathrm{[}G\mathrm{]}$. Define the Stickelberger element $\theta \mathrm{\in }\mathrm{Q}\mathit{}\mathrm{[}G\mathrm{]}$ by

$\theta ={\displaystyle \frac{1}{m}}{\displaystyle \sum _{1\le a\le m,(a,m)=1}}a{\sigma }_a^{-1},$

and take $\beta \mathrm{\in }\mathrm{Z}\mathit{}\mathrm{[}G\mathrm{]}$ such that $\beta \mathit{}\theta \mathrm{\in }\mathrm{Z}\mathit{}\mathrm{[}G\mathrm{]}$ as well. Then $\beta \mathit{}\theta$ is an annihilator for the ideal class group of $\mathrm{Q}\mathit{}\mathrm{(}{\zeta }_m\mathrm{)}$.

Note that $\theta$ itself need not be an annihilator, just that any multiple of it in $\mathbb{Z}[G]$ is.

This result allows for the most basic between the (otherwise hard to determine) of a cyclotomic ideal class group and its of annihilators. For an application of Stickelberger’s theorem, see Herbrand’s theorem.