Stickelberger's theorem

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

$\theta ={\frac {1}{m}}\sum _{1\leq a\leq m,(a,m)=1}a\sigma _{a}^{-1}.$

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 $\mathbb {Q} (\zeta _{m})$, as Galois module.

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

External link

• PlanetMath page