Stickelberger's theorem

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In mathematics, Stickelberger's theorem is a result of algebraic number theory, which gives some information about the Galois module structure of class groups of cyclotomic fields. It is due to Ludwig Stickelberger (1850-1936).

Theorem (Stickelberger) Let $\mathbb{Q}(\zeta_m)$ be a cyclotomic field extension of $\mathbb{Q}$ with Galois group $G = \{\sigma_a | a \in (\mathbb Z / m\mathbb Z)^*\}$ , and consider the group ring $\mathbb{Q}[G]$ . Define the Stickelberger element $\theta \in \mathbb{Q}[G]$ by

$\theta = \frac 1 m \sum_{a \in (\mathbb Z / m\mathbb Z)^*} a \sigma_a^{-1}.$

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