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

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 ${\mathcal{O}}_L$.