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Computation of Hopf Galois structures on separable extensions and classification of those for degree twice an odd prime power

Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain

E-mail Address: teresa.crespo@ub.edu

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and

Marta Salguero

Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain

E-mail Address: msalguga11@alumnes.ub.edu

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https://doi.org/10.1142/S0219498821500493Cited by:1 (Source: Crossref)

Abstract

A Hopf Galois structure on a finite field extension $L / K$ $L / K$ is a pair $(H, μ)$ $(H, μ)$ , where $H$ $H$ is a finite cocommutative $K$ $K$ -Hopf algebra and $μ$ $μ$ a Hopf action. In this paper, we present a program written in the computational algebra system Magma which gives all Hopf Galois structures on separable field extensions of a given degree and several properties of those. We show a table which summarizes the program results. Besides, for separable field extensions of degree $2 p^{n}$ $2 p^{n}$ , with $p$ $p$ an odd prime number, we prove that the occurrence of some type of Hopf Galois structure may either imply or exclude the occurrence of some other type. In particular, for separable field extensions of degree $2 p^{2}$ $2 p^{2}$ , we determine exactly the possible sets of Hopf Galois structure types.

Communicated by J. Gómez-Torrecillas

Keywords:

AMSC: 12F10, 16T05, 33F10, 20B05

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