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arXiv · 1407.4175

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

Cindy·Tsang
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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 $\mathcal{O}_KG$. We show that for $G$ abelian and under suitable assumptions, the set of all such classes is a subgroup of $\mbox{Cl}(\mathcal{O}_KG)$.

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Cindy, Tsang. 2015-06-09. On the Galois module structure of the square root of the inverse different in abelian extensions. https://doi.org/10.1016/j.jnt.2015.09.010

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