On the Galois module structure of the square root of the inverse different in abelian extensions

Abstract

Let $K$ be a number field with ring of integers ${\mathcal{O}}_K$ and $G$ a finite group of odd order. If $K_h$ is a weakly ramified $G$-Galois $K$-algebra, then its square root $A_h$ of the inverse different is a locally free ${\mathcal{O}}_{K}G$-module and hence determines a class in the locally free class group $\mbox{Cl}(\mathcal{O}_KG)$ of ${\mathcal{O}}_{K}G$. We show that for $G$ abelian and under suitable assumptions, the set of all such classes is a subgroup of $\mbox{Cl}(\mathcal{O}_KG)$.

version 3; we improved the statements of the theorems and simplified their proofs significantly

References in corpus (2)

• Self-Dual Integral Normal Bases and Galois Module Structure
• Explicit Construction of Self-Dual Integral Normal Bases for the Square-Root of the Inverse Different

Cited by in corpus (5)

• On the relative Galois module structure of rings of integers in tame extensions
• On the realizable classes of the square root of the inverse different in the unitary class group
• On the self-duality of rings of integers in tame and abelian extensions
• Galois module structure of the square root of the inverse different over maximal orders
• Realizable classes and embedding problems