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

@inproceedings{GilMuoz2021HopfGM, title={Hopf Galois module structure of quartic Galois extensions of \$\mathbb\{Q\}\$}, author={Daniel Gil-Mu{\~n}oz and A. Rio}, year={2021} }

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 AH in H . For the classical Galois structure Hc, we know by Leopoldt’s theorem that OL is AHc -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

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