Stickelberger Relation

Let $P$ be a Prime Ideal in $D_m$ not containing $m$. Then

$$(\Phi(P))=P^{\sum t\sigma_t^{-1}},$$

where the sum is over all $1\leq t<m$ which are Relatively Prime to $m$. Here $D_m$ is the Ring of integers in $\Bbb{Q}(\zeta_m)$, $\Phi(P)=g(P)^m$, and other quantities are defined by Ireland and Rosen (1990).

See also Prime Ideal

References

Ireland, K. and Rosen, M. ``The Stickelberger Relation and the Eisenstein Reciprocity Law.'' Ch. 14 in A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, pp. 203-227, 1990.