Hopf-Galois module structure of quartic Galois extensions of $\mathbb{Q}$

TL;DR

This paper investigates the module structure of the ring of integers in quartic Galois extensions of rationals under various Hopf-Galois structures, linking freeness to solutions of generalized Pell equations.

Contribution

It extends classical results by analyzing non-classical Hopf-Galois structures and relating module freeness to Pell equation solvability in quartic Galois extensions.

Findings

Freeness depends on solvability of generalized Pell equations.

Classical structure always yields free modules (Leopoldt's theorem).

Non-classical structures' freeness characterized by Pell equation solutions.

Abstract

Given a quartic Galois extension $L/\mathbb{Q}$ of number fields and a Hopf-Galois structure $H$ on $L/\mathbb{Q}$, we study the freeness of the ring of integers $\mathcal{O}_L$ as module over the associated order $\mathfrak{A}_H$ in $H$. For the classical Galois structure $H_c$, we know by Leopoldt's theorem that $\mathcal{O}_L$ is $\mathfrak{A}_{H_c}$-free. If $L/\mathbb{Q}$ is cyclic, it admits a unique non-classical Hopf-Galois structure, whereas if it is biquadratic, it admits three such Hopf-Galois structures. In both cases, we obtain that freeness depends on the solvability in $\mathbb{Z}$ of certain generalized Pell equations. We shall translate some results on Pell equations into results on the $\mathfrak{A}_H$-freeness of $\mathcal{O}_L$.