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# Stickelberger’s theorem

###### Theorem 1 (Stickelberger).

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

$\displaystyle\theta=\frac{1}{m}\sum_{{1\leq a\leq m,(a,m)=1}}a\sigma_{a}^{{-1}},$ |

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

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 connections between the (otherwise hard to determine) structure of a cyclotomic ideal class group and its collection of annihilators. For an application of Stickelberger’s theorem, see Herbrand’s theorem.

## Mathematics Subject Classification

11R29*no label found*

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