Hopf--Galois structures of cyclic type on parallel extensions of prime power degree

2025/10/16 by Darlington, Andrew, Tsang, Cindy
#FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2510.14473

Abstract

Let L/K be any finite separable extension with normal closure \widetildeL/K. An extension L'/K is said to be parallel to L/K if L' is an intermediate field of \widetildeL/K with [L':K]=[L:K]. We study the following question -- Given that L/K admits a Hopf--Galois structure of type N, does it imply that every extension parallel to L/K also admits a Hopf--Galois structure of type N? We completely solve this problem when the degree [L:K] is a prime power and the type N is cyclic. Our approach is group-theoretic and uses the work of Greither--Pareigis and Byott.

Citations

• Hopf-Galois structures on parallel extensions
• Automatic realization of Hopf Galois structures

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