Theorem 1 (Stickelberger)   Let $L=\mathbb{Q}(\zeta_m)$ be a cyclotomic field extension of $\Q$ with Galois group $G=\{\sigma_a\}_{a\in(\Z/m\Z)^\times}$ , and consider the group ring $\Q[G]$ . Define the Stickelberger element $\theta\in\Q[G]$ by

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

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