In this setting one can consider a separable extension (which may not be Galois in the usual sense) as being Hopf-Galois where instead of a group acting on the extension, one has a Hopf algebra which acts. Note that a classical Galois extension L/K with G=Gal(L/K) is also Hopf-Galois in a very natural way under the action of the Hopf algebra H=K[G].

The work of Greither and Pareigis showed how to classify and enumerate such extensions by looking at certain permutation groups.

In broad terms Hopf-Galois theory can be viewed as an aspect of Galois Module Theory.

Publications

• Books
  • Hopf Algebras and Galois Module Theory (with Childs L, Greither C, Keating K, Koch A,Truman PJ, Underwood R.) American Mathematical Society - Mathematical Surveys and Monographs - Volume 260.

• Journal Articles
  • [17] Hopf Forms and Hopf-Galois Theory - New York J. Math. 31 (2025) (with R. Underwood)

  • [16] Mutually Normalizing Regular Permutation Groups and Zappa-Szep Extensions of the Holomorph Rocky Mountain Journal of Mathematics, Volume 52, Number 5, 567--598, 2022

  • [15] Enumerating Dihedral Hopf-Galois Structures Acting on Dihedral Extensions, J. Algebra Volume 542, 15 January 2020, Pages 93-115

  • [14] The Structure of Hopf Algebras Acting on Dihedral Extensions (with A. Koch, P.J. Truman, R. Underwood), in J. Feldvoss et al. (eds) Advances in Algebra, Springer Proceedings in Mathematics & Statistics 277 (2019)

  • [13] Characteristic Subgroups Lattices and Hopf-Galois Structures, International Journal of Algebra and Computation, Vol. 29, No. 02, pp. 391-405 (2019)

  • [12] Isomorphism problems for Hopf-Galois structures on separable field extensions - Journal of Pure and Applied Algebra, Vol. 223, Issue 5, pp. 2230-2245, May 2019 (with A. Koch, P.J. Truman, R. Underwood)

  • [11] Normality and Short Exact Sequences of Hopf-Galois Structures, Communications in Algebra,Vol. 47, No. 5, pp. 2086-2101. 2018 (with A. Koch, P.J. Truman, R. Underwood)

  • [10] A Class of Profinite Hopf-Galois Extensions Over Q Communications in Algebra - 2017 - DOI: 10.1080/00927872.2017.1384001

  • [9] Hopf-Galois Structures Arising From Groups With Unique Subgroup of Order p Algebra & Number Theory 10-1 (2016), 37--59. DOI: 10.2140/ant.2016.10-1

  • [8] Multiple Holomorphs of Dihedral and Quaternionic Groups, Communications in Algebra 43 (2015) 4290-4304

  • [7] Regular Permutation Groups of Order mp and Hopf Galois Structures, Algebra & Number Theory 7-9 (2013), 2203--2240. DOI 10.2140/ant.2013.7.2203

  • [6] When Abelian = Hausdorf, College Math Journal 43 (3), (2012), 213--215.

  • [5] Groups of Order 4p, Twisted Wreath Products and Hopf-Galois Theory J. Algebra, 314 (2007), 42-74.
    Errata:
    • P.72, just before Prop. 6.9, should be \(P(N)\neq P_1\) since \(P_1\) is the p-Sylow subgroup of \(\lambda(\Gamma)\) but *not* \(\rho(\Gamma)\)
    • Corollary 3.12: .. and \(|R(\Gamma,[D_{2p}])|+|R(\Gamma,[Q_{p}])|\) is divisible by 4.
  • [4]Cyclotomic Swan Subgroups and Primitive Roots (with D. Replogle), Finite Fields and Their Applications,11 (2005),655-666

  • [3]Computation of Several Cyclotomic Swan Subgroups (with D. Replogle), Math. Comp., 71 (2002),343-348

  • [2] Classification of the Hopf Galois Structures on Prime Power Radical Extensions, J. Algebra, 207 (1998), 525-546

  • [1] Group Rings and Hopf Galois Theory in Maple in Maple V: Mathematics and Its Application, Proceedings of the the Maple Summer Workshop and Symposium, Rensselear Polytechnic Institute, Troy, New York, August 9-13, 1994, Robert J. Lopez, Editor, Birkhaü ser Boston, 1994.

Presentations/Conferences

• 2024
  • Hopf Algebras and Galois Module Theory - University of Omaha Nebraska, May 28- 31 2024
    • Isomorphic Holomorphs
• 2023
  • Hopf Algebras and Galois Module Theory - University of Omaha Nebraska, May 29- June 2, 2023
    • Beyond Isomorphic Holomorphs
• 2022
  • Hopf Algebras and Galois Module Theory - University of Omaha Nebraska, May 30- June 3, 2022
    • Normalizing Graphs of Regular Permutation Groups
• 2021
  • Hopf Algebras and Galois Module Theory - An Online Conference, May 24-27, 2021
    • Isomorphic Holomorphs and Bi-Skew Braces
• 2020
  • Hopf Algebras and Galois Module Theory - An Online Conference, May 25-28, 2020
    • Mutually Normalizing Regular Permutation Groups and Zappa-Szep Extensions of the Holomorph
• 2019
  • Hopf Algebras and Galois Module Theory - University of Nebraska at Omaha, May 27-31, 2019
    • Enumerating Hopf-Galois Structures on Dihedral Extensions
    • Multiple Holomorphs and Isomorphism Classes of Hopf-Galois Structures on Dihedral Extensions
• 2018
  • Hopf Algebras and Galois Module Theory - University of Nebraska at Omaha, May 21-25, 2018
    • Characteristic Subgroup Lattices and Hopf-Galois Structures
    • Multiple Holomorphs and Hopf-Galois Structures
• 2017
  • Hopf Algebras and Galois Module Theory - University of Nebraska at Omaha, May 22-26, 2017
    • Normality and Short Exact Sequences of Hopf-Galois Structures
    • A Class of Profinite Hopf-Galois Extensions over Q
• 2016
  • Hopf Algebras and Galois Module Theory - University of Nebraska at Omaha, May 23-30, 2016
    • Regular Permutation Groups and Their Centralizers and Normalizers
    • Regular Permutation Groups and Their Centralizers and Normalizers II
    • Hopf Galois Structures on Degree mp Extensions
• 2015
  • Tufts University - Algebra and Geometry Seminar - March 25, 2015
  • Presented 'Multiple Holomorphs of Dihedral and Quaternionic Groups'
• 2014
  • Southern Regional Algebra Conference - Auburn University Montgomery - April 25-27, 2014
  • Presented 'Multiple Holomorphs of Dihedral and Quaternionic Groups'
• 2013
  • State University of New York At Albany - Algebra and Topology Seminar - April 11, 2013
  • Presented 'Regular Permutation Groups of Order mp and Hopf Galois Structures'
• 2012
  • AMS Spring South Eastern Sectional Meeting Special Section on Hopf Algebras and Galois Module Theory - March 10, 2012
  • Presented 'Hopf Galois Structures Arising from Mutually Normalizing Permutation Groups'
• 2010
  • U. Mass Boston - Mathematics Seminar - April 5, 2010
  • Presented 'Regular Permutation Groups of Order mp'
• 2009
  • AMS Spring Eastern Sectional Meeting - Special Section on Hopf Algebras and Galois Module Theory
  • University of North Carolina Raleigh - April 2009
  • Presented 'Regular Permutation Groups of Order mp'
• 2006
  • AMS Spring Eastern Sectional Meeting - Special Section on Hopf Algebras and Galois Module Theory
  • Univ. of New Hampshire - April 2006
  • Co-Organizer (with R.Underwood) of this special section
  • Presented 'Groups of Order 4p, Twisted Wreath Products and Hopf Galois Theory'
• 2003
  • AMS Spring Eastern Sectional Meeting - Special Section on Hopf Algebras and Galois Module Theory
  • April 13 - Counting Hopf Galois Structures on Dihedral and Quaternionic Extensions
• 2002
  • Boston University Electro-Physics Brown Bag Lunch
    • September 23 - Hopf Algebras - Basic Notions
    • October 3 - Hopf Algebras - Radical Notions
• 1999
  • Boston University Algebra Seminar
    • March 22 - Hopf Galois Theory for Fields
• 1996
  • Capital Region Algebra Number Theory Seminar, at Union College
    • May 15 - Hopf Galois Structures on Radical Extensions
• 1994
  • Maple Summer Workshop and Symposium, Rensselear Polytechnic Institute, Troy, New York
    • August 10 - Group Rings and Hopf Galois Theory in Maple