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The Structure of Hopf Algebras Acting on Dihedral Extensions

  • Alan Koch
  • Timothy Kohl
  • Paul J. Truman
  • Robert UnderwoodEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 277)

Abstract

We discuss isomorphism questions concerning the Hopf algebras that yield Hopf–Galois structures for a fixed separable field extension L/K. We study in detail the case where L/K is Galois with dihedral group \(D_p\), \(p\ge 3\) prime and give explicit descriptions of the Hopf algebras which act on L/K. We also determine when two such Hopf algebras are isomorphic, either as Hopf algebras or as algebras. For the case \(p=3\) and a chosen L/K, we give the Wedderburn–Artin decompositions of the Hopf algebras.

Keywords

Hopf–Galois extension Dihedral extension Wedderburn–Artin decomposition 

Notes

Acknowledgements

The authors would like to thank the referee for comments and suggestions which improved the exposition and content of this paper.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Alan Koch
    • 1
  • Timothy Kohl
    • 2
  • Paul J. Truman
    • 3
  • Robert Underwood
    • 4
    Email author
  1. 1.Department of MathematicsAgnes Scott CollegeDecaturUSA
  2. 2.Department of Mathematics and StatisticsBoston UniversityBostonUSA
  3. 3.School of Computing and MathematicsKeele UniversityStaffordshireUK
  4. 4.Department of Mathematics and Computer ScienceAuburn University at MontgomeryMontgomeryUSA

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