Construction of a $p$-extension of number fields whose unit group has prescribed Galois module structure

TL;DR

This paper constructs specific number field extensions with a prescribed Galois module structure for their unit groups, advancing the understanding of algebraic number theory and Galois module theory.

Contribution

It provides a method to explicitly construct Galois extensions of number fields with unit groups having a specified module structure over the group ring.

Findings

Constructed Galois extensions with prescribed unit group structures

Demonstrated control over the Galois module structure of units

Extended the theory of Galois modules in number fields

Abstract

Let $G$ be a finite $p$-group. We construct a $G$-extension $K/k$ of number fields such that the $p$-adic completion of the unit group of $K$ has a prescribed $\mathbb{Z}_p[G]$-module structure, up to free direct summands.