STNB2026 (39th edition)
Hopf-Galois extensions in number theory
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Resumen
It was at the end of the eighties when Childs suggested that Hopf-Galois theory could be applied in order to generalize Galois module theory, which consists in the study of the module structure of the ring of algebraic integers over the Galois group algebra in a Galois extension of number or p-adic fields. This idea led to the development of Hopf-Galois module theory, which considers instead Hopf-Galois extensions of number or p-adic fields, and the ground ring for the module structure of the algebraic integers is an object depending on a Hopf-Galois structure on the extension. This problem contains the one corresponding to classical Galois module theory and has been useful to broaden the information provided by the latter. In this lecture, we shall present the main research directions in this topic. In particular, the positive results obtained so far will illustrate how the behavior of the module structure of the ring of integers depends on arithmetic invariants on the extension.
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