Canonical Nonclassical Hopf-Galois Module Structure of Nonabelian Galois Extensions

TL;DR

This paper investigates the module structure of fractional ideals in nonabelian Galois extensions, establishing conditions under which these ideals are free over associated orders in both classical and nonclassical Hopf-Galois structures.

Contribution

It proves the equivalence of freeness over classical and nonclassical Hopf orders for fractional ideals in nonabelian Galois extensions.

Findings

Fractional ideals are free over their associated orders in classical Hopf-Galois structures.

Freeness over classical and nonclassical structures are equivalent for these ideals.

Results apply to local and global fields in characteristic 0 or p.

Abstract

Let $L/K$ be a finite Galois extension of local or global fields in characteristic $0$ or $p$ with nonabelian Galois group $G$, and let ${\mathfrak B}$ be a $G$-stable fractional ideal of $L$. We show that ${\mathfrak B}$ is free over its associated order in $K[G]$ if and only if it is free over its associated order in the Hopf algebra giving the canonical nonclassical Hopf-Galois structure on the extension.