Nilpotent and abelian Hopf-Galois structures on field extensions

2012/10/05 by Byott, Nigel P.
#12F10 #16T05 #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1210.1693

Abstract

Let L/K be a finite Galois extension of fields with group Γ. When Γ is nilpotent, we show that the problem of enumerating all nilpotent Hopf-Galois structures on L/K can be reduced to the corresponding problem for the Sylow subgroups of Γ. We use this to enumerate all nilpotent (resp. abelian) Hopf-Galois structures on a cyclic extension of arbitrary finite degree. When Γ is abelian, we give conditions under which every abelian Hopf-Galois structure on L/K has type Γ. We also give a criterion on n such that every Hopf-Galois structure on a cyclic extension of degree n has cyclic type.

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