2020 Computation of Hopf Galois structures on low degree separable extensions and classification of those for degrees $p^2$ and $2p$
Teresa Crespo, Marta Salguero
Publ. Mat. 64(1): 121-141 (2020). DOI: 10.5565/PUBLMAT6412005

Abstract

A Hopf Galois structure on a finite field extension $L/K$ is a pair $(H,\mu)$, where $H$ is a finite cocommutative $K$-Hopf algebra and $\mu$ 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 degree up to eleven and several properties of those. Besides, we exhibit several results on Hopf Galois structures inspired by the program output. We prove that if $(H,\mu)$ is an almost classically Hopf Galois structure, then it is the unique Hopf Galois structure with underlying Hopf algebra $H$ up to isomorphism. For $p$ an odd prime, we prove that a separable extension of degree $p^2$ may have only one type of Hopf Galois structure and determine those of cyclic type; we determine as well the Hopf Galois structures on separable extensions of degree $2p$. We highlight the richness of the results obtained for extensions of degree $8$ by computing an explicit example and presenting some tables which summarize these results.

Funding Statement

Both authors acknowledge support by grant MTM2015-66716-P (MINECO/FEDER, UE)

Citation

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Teresa Crespo. Marta Salguero. "Computation of Hopf Galois structures on low degree separable extensions and classification of those for degrees $p^2$ and $2p$." Publ. Mat. 64 (1) 121 - 141, 2020. https://doi.org/10.5565/PUBLMAT6412005

Information

Received: 26 February 2018; Revised: 27 June 2018; Published: 2020
First available in Project Euclid: 3 January 2020

zbMATH: 07173899
MathSciNet: MR4047559
Digital Object Identifier: 10.5565/PUBLMAT6412005

Subjects:
Primary: 12F10
Secondary: 16T05 , 20B05 , 33F10

Keywords: computational algebra system Magma , Galois theory , Hopf algebra

Rights: Copyright © 2020 Universitat Autònoma de Barcelona, Departament de Matemàtiques

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Vol.64 • No. 1 • 2020
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