Isomorphism problems for Hopf-Galois structures on separable field extensions

Koch, Alan; Kohl, Timothy; Truman, Paul J.; Underwood, Robert

Abstract

Let L=K be a finite separable extension of fields whose Galois closure E=K has Galois group G. Greither and Pareigis use Galois descent to show that a Hopf algebra giving a Hopf-Galois structure on L=K has the form E[N]G for some group N of order [L : K]. We formulate criteria for two such Hopf algebras to be isomorphic as Hopf algebras, and provide a variety of examples. In the case that the Hopf algebras in question are commutative, we also determine criteria for them to be isomorphic as K-algebras. By applying our results, we complete a detailed analysis of the distinct Hopf algebras and K-algebras that appear in the classification of Hopf-Galois structures on a cyclic extension of degree pn, for p an odd prime number.

Citation

Koch, A., Kohl, T., Truman, P. J., & Underwood, R. (2019). Isomorphism problems for Hopf-Galois structures on separable field extensions. Journal of Pure and Applied Algebra, 2230-2245. https://doi.org/10.1016/j.jpaa.2018.07.014

• Acceptance Date: Jun 26, 2018
• Publication Date: May 1, 2019
• Journal: Journal of Pure and Applied Algebra
• Print ISSN: 0022-4049
• Publisher: Elsevier
• Pages: 2230-2245
• DOI: https://doi.org/10.1016/j.jpaa.2018.07.014
• Keywords: Hopf-Galois extension, Greither-Pareigis theory, Galois descent, 2000 MSC: 16T05
• Publisher URL: http://doi.org/10.1016/j.jpaa.2018.07.014