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 $\mathcal{ O}_L$.
Categories: math.NT