# On the self-duality of rings of integers in tame and abelian extensions

## TL;DR

This paper explores the self-duality properties of rings of integers in tame, abelian Galois extensions of number fields, focusing on the relationships among various ideal classes and their duals.

## Contribution

It establishes new connections between the classes of the ring of integers, inverse different, and square root of inverse different in abelian extensions, highlighting their self-duality properties.

## Key findings

- A_{L/K}^2 = \, ext{inverse different} = \, ext{dual of} \, ext{ring of integers}
- A_{L/K} is self-dual with respect to the trace pairing
- Relationships among ideal classes are characterized in abelian Galois extensions

## Abstract

Let $L/K$ be a tame and Galois extension of number fields with group $G$. It is well-known that any ambiguous ideal in $L$ is locally free over $\mathcal{O}_KG$ (of rank one), and so it defines a class in the locally free class group of $\mathcal{O}_KG$, where $\mathcal{O}_K$ denotes the ring of integers of $K$. In this paper, we shall study the relationship among the classes arising from the ring of integers $\mathcal{O}_L$ of $L$, the inverse different $\mathfrak{D}_{L/K}^{-1}$ of $L/K$, and the square root of the inverse different $A_{L/K}$ of $L/K$ (if it exists), in the case that $G$ is abelian. They are naturally related because $A_{L/K}^2 = \mathfrak{D}_{L/K}^{-1} = \mathcal{O}_L^*$, and $A_{L/K}$ is special because $A_{L/K} = A_{L/K}^*$, where $*$ denotes dual with respect to the trace of $L/K$.