Abelian Hopf Galois structures on prime-power Galois field extensions

By S. C. Featherstonhaugh, Andrea Caranti, Lindsay N. Childs. 2012

51 citations recorded by the source corpus.

The main theorem of this paper is that if (N, +) is a finite abelian p-group of p-rank m where m + 1 < p, then every regular abelian subgroup of the holomorph of N is isomorphic to N . The proof utilizes a connection, observed in [CDVS06], between regular abelian subgroups of the holomorph of N and nilpotent ring structures on (N, +). Examples are given that limit possible generalizations of the theorem. The primary application of the theorem is to Hopf Galois extensions of fields. Let L|K be a Galois extension of fields with abelian Galois group G. If also L|K is HHopf Galois where the K-Hopf algebra H has associated group N with N as above, then N is isomorphic to G.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/4d695af4b07a62df03f5422b3125bab62814440a

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Citations in the corpus

Bi-skew braces and Hopf Galois structures (2019, unknown): “Generalizing [13] for (B,+) a radical ring, Bachiller [3] proved that if m+2 ≤ p, then for every b in B, the order of b in (B, ◦) is equal to the order of b in (B,+).”
Extensions, matched products, and simple braces (2015, background): “Besides its relation with theoretical physics, this class of solutions is important for its connections with many branches of algebra, for instance: semigroups of I-type and Bieberbach groups [12], bijective 1-cocycles [10], radical rings [17], triply factorized groups [19], construction of semisimp”
Counterexample to a conjecture about braces (2015, methodology): “In [13], the proof is in terms of radical rings, but remember that two-sided braces and radical rings are equivalent by [8, Proposition 1].”
Solutions of the Yang-Baxter equation associated with a left brace (2015, background): “the Yang-Baxter equation in physics, but also for its connection with other topics in mathematics of recent interest: semigroups of I-type and Bieberbach groups [16], bijective 1-cocycles [11], radical rings [23], triply factorized groups [25], construction of semisimple minimal triangular Hopf alge”