On the Galois module structure of the square root of the inverse different in abelian extensions
Copyright policy )On the Galois module structure of the square root of the inverse different in abelian extensions
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 $\mbox{Cl}(\mathcal{O}_KG)$.
version 3; we improved the statements of the theorems and simplified their proofs significantly
- University of California, Santa Barbara United States
weakly ramified, Mathematics - Number Theory, square root of the inverse different, FOS: Mathematics, Integral representations related to algebraic numbers; Galois module structure of rings of integers, Galois module, Number Theory (math.NT), realizable classes, Algebraic numbers; rings of algebraic integers
weakly ramified, Mathematics - Number Theory, square root of the inverse different, FOS: Mathematics, Integral representations related to algebraic numbers; Galois module structure of rings of integers, Galois module, Number Theory (math.NT), realizable classes, Algebraic numbers; rings of algebraic integers
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