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Isomorphism problems for Hopf-Galois structures on separable field extensions (2019)
Journal Article
Koch, A., Kohl, T., Truman, P. J., & Underwood, R. (2019). Isomorphism problems for Hopf-Galois structures on separable field extensions. Journal of Pure and Applied Algebra, 2230-2245. https://doi.org/10.1016/j.jpaa.2018.07.014

Let L=K be a finite separable extension of fields whose Galois closure E=K has Galois group G. Greither and Pareigis use Galois descent to show that a Hopf algebra giving a Hopf-Galois structure on L=K has the form E[N]G for some group N of order [L... Read More about Isomorphism problems for Hopf-Galois structures on separable field extensions.

The structure of Hopf algebras giving Hopf-Galois structures on Quaternionic extensions (2019)
Journal Article
Truman, P., & Taylor, S. (2019). The structure of Hopf algebras giving Hopf-Galois structures on Quaternionic extensions

Let L/F be a Galois extension of fields with Galois group isomorphic to the quaternion group of order 8. We describe all of the Hopf-Galois structures admitted by L/F, and determine which of the Hopf algebras that appear are isomorphic as Hopf algebr... Read More about The structure of Hopf algebras giving Hopf-Galois structures on Quaternionic extensions.

The Structure of Hopf Algebras Acting on Dihedral Extensions (2019)
Book Chapter
Koch, A., Kohl, T., Truman, P. J., & Underwood, R. (2019). The Structure of Hopf Algebras Acting on Dihedral Extensions. In Advances in Algebra. SRAC 2017 (201-218)

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 Dp, p=3 prime and give explicit description... Read More about The Structure of Hopf Algebras Acting on Dihedral Extensions.

Normality and Short Exact Sequences of Hopf-Galois Structures (2019)
Journal Article
Koch, A., Kohl, T., Truman, P. J., & Underwood, R. (2019). Normality and Short Exact Sequences of Hopf-Galois Structures. Communications in Algebra, 2086-2101. https://doi.org/10.1080/00927872.2018.1529237

Every Hopf-Galois structure on a finite Galois extension K/k where G = Gal(K/k) corresponds uniquely to a regular subgroup N = B = Perm(G), normalized by ?(G) = B, in accordance with a theorem of Greither and Pareigis. The resulting Hopf algebra whic... Read More about Normality and Short Exact Sequences of Hopf-Galois Structures.