Hopf-Galois module structure of quartic Galois extensions of Q

Abstrakt

Given a quartic Galois extension L/Q of number fields and a Hopf-Galois structure H on L/Q, we study the freeness of the ring of integers OL as module over the associated order %H in H. For the classical Galois structure Hc, we know by Leopoldt's theorem that OL is %Hc-free.

If L/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 Z of certain generalized Pell equations.

We shall translate some results on Pell equations into results on the %H-freeness of OL. (C) 2022 The Author(s). Published by Elsevier B.V.

Klíčová slova

• Hopf-Galois structure
• Freeness over the associated order