Let $F/\Q$ be a totally real abelian number field. By the Kronecker-Weber theorem, there exists an $m$ such that $F\subset \Q(\zeta_m)$ . Let $G$ be the Galois group of the extension $F/Q$ . Let $\mc{O}_F^\times$ denote the group of units in the ring of integers of $F$ , let $C$ be the subgroup of $\mc{O}_F^\times$ consisting of units $\eta$ of the form

This is one of the most sophisticated results concerning the annihilators of an ideal class group. It is a direct, but more complicated, version of Stickelberger's theorem, applied to totally real fields (for which Stickelberger's theorem gives no information).