Galois module structure of local unit groups

Abstract

We study the groups $U_i$ in the unit filtration of a finite abelian extension $K$ of ${\mathbb{Q}}_p$ for an odd prime $p$. We determine explicit generators of the $U_i$ as modules over the ${\mathbb{Z}}_p$-group ring of $Gal\left(K∕{\mathbb{Q}}_p\right)$. We work in eigenspaces for powers of the Teichmüller character, first at the level of the field of norms for the extension of $K$ by $p$-power roots of unity and then at the level of $K$.