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Title: On Hopf-Galois structures and skew braces of order p³
Author: Nejabati Zenouz, Kayvan
ISNI:       0000 0004 7226 1920
Awarding Body: University of Exeter
Current Institution: University of Exeter
Date of Award: 2018
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The concept of Hopf-Galois extensions was introduced by S. Chase and M. Sweedler in 1969 and provides a generalisation of classical Galois theory. Later, Hopf-Galois theory for separable extensions of fields was studied by C. Greither and B. Pareigis. They showed how to recast the problem of classifying all Hopf-Galois structures on a finite separable extension of fields as a problem in group theory. Many major advances relating to the classification of Hopf-Galois structures were made by N. Byott, S. Carnahan, L. Childs, and T. Kohl. On the other hand, and seemingly unrelated to Hopf-Galois theory, in 1992 V. Drinfeld formulated a number of problems in quantum group theory. In particular, he suggested considering set-theoretic solutions of the Yang-Baxter equation. Later, W. Rump introduced braces as a tool to study non-degenerate involutive set-theoretic solutions, and through the efforts of D. Bachiller, F. Ced'o, E. Jespers, and J. Okni'nski the classification of these solutions was reduced to that of braces. Recently, skew braces were introduced by L. Guarnieri and L. Vendramin in order to study the non-degenerate (not necessarily involutive) set-theoretic solutions. Additionally, a fruitful discovery, initially noticed by D. Bachiller, revealed a connection between Hopf-Galois theory and skew braces, which linked the classification of Hopf-Galois structures to that of skew braces. Currently, the classification of Hopf-Galois structures and skew braces of a given order remains among important topics of research. In this thesis, as our main results, we determine all Hopf-Galois structures on Galois extensions of fields of degree p^3, and at the same time we provide a complete classification of all skew braces of order p^3, for a prime number p. These findings hence offer applications to Galois module theory in number theory on the one hand, and to the study of the solutions of the quantum Yang-Baxter equation in mathematical physics on the other hand.
Supervisor: Byott, Nigel Sponsor: EPSRC
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Hopf-Galois structures ; Skew barces