On the rank of the first radical layer of a $p$-class group of an algebraic number field

Abstract

Let $p$ be a prime number. Let $M$ be a finite Galois extension of a finite algebraic number field $k$. Suppose that $M$ contains a primitive $p$th root of unity and that the $p$-Sylow subgroup of the Galois group $G=Gal(M/k)$ is normal. Let $K$ be the intermediate field corresponding to the $p$-Sylow subgroup. Let ${\frak g}=Gal(K/k)$. The $p$-class group ${\cal C}$ of $M$ is a module over the group ring ${\bf Z}_p G$, where ${\bf Z}_p$ is the ring of $p$-adic integers. Let $J$ be the Jacobson radical of ${\bf Z}_p G$. ${\cal C}/J {\cal C}$ is a module over a semisimple artinian ring ${\bf F}_p {\frak g}$. We study multiplicity of an irreducible representation $\Phi$ apperaring in ${\cal C}/J{\cal C}$ and prove a formula giving this multiplicity partially. As application to this formula, we study a cyclotomic field $M$ such that the minus part of ${\cal C}$ is cyclic as a ${\bf Z}_p G$-module and a CM-field $M$ such that the plus part of ${\cal C}$ vanishes for odd $p$.

To show the formula, we apply theory of central extensions of algebraic number field andstudy global and local Kummer duality between the genus group and the Kummer radical for the genus field with respect to $M/K$.