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

TL;DR

This paper constructs specific cyclic p-extensions of number fields where the unit group's Galois module structure can be prescribed, demonstrating the realization of all torsion-free modules as unit groups in such extensions.

Contribution

It shows that any torsion-free finitely generated module over the group ring can be realized as the unit group in a cyclic p-extension of number fields.

Findings

Realization of all torsion-free modules as unit groups.

Construction of cyclic p-extensions with prescribed Galois module structure.

Extension of the understanding of unit groups in number field extensions.

Abstract

For any finite cyclic $p$-group $G$, we will show that every $\mathbb{Z}_p$-torsion free finitely generated $\mathbb{Z}_p[G]$-module appears as $\mathcal{O}_K^\times\otimes_{\mathbb{Z}}\mathbb{Z}_p$ up to $\mathbb{Z}_p[G]$-free direct summands for a certain $G$-extension $K/k$ of number fields.