Minimal Hopf-Galois Structures on Separable Field Extensions

2020/11/15 by Ezome, Tony, Greither, Cornelius
#12F10 #FOS: Mathematics #Number Theory (math.NT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2011.07578

Abstract

In Hopf-Galois theory, every H-Hopf-Galois structure on a field extension K/k gives rise to an injective map F from the set of k-sub-Hopf algebras of H into the intermediate fields of K/k. Recent papers on the failure of the surjectivity of F reveal that there exist many Hopf-Galois structures for which there are many more subfields than sub-Hopf algebras. This paper surveys and illustrates group-theoretical methods to determine H-Hopf-Galois structures on finite separable extensions in the extreme situation when H has only two sub-Hopf algebras.

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