On counting rings of integers as Galois modules

TL;DR

This paper investigates the asymptotic distribution of tamely ramified G-extensions of a number field K, focusing on the Galois module structure of their rings of integers, providing insights into their counting behavior.

Contribution

It introduces a new approach to counting Galois extensions with specified Galois module structures, advancing understanding of their asymptotic properties.

Findings

Derived asymptotic formulas for the number of such extensions

Identified conditions affecting the distribution of Galois modules

Provided new insights into the structure of rings of integers as Galois modules

Abstract

Let $K$ be a number field and $G$ a finite abelian group. We study the asymptotic behaviour of the number of tamely ramified $G$-extensions of $K$ with ring of integers of fixed realisable class as a Galois module.