Robert Underwood — Page 3

arXiv2017

Isomorphism problems for Hopf-Galois structures on separable field extensions

Let $ L/K $ be a finite separable extension of fields whose Galois closure $ E/K $ has group $ G $. Greither and Pareigis have used 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 $ such that $ |N|=[L:K] $. We formulate criteria for two such Hopf algebras to be isomorphic as Hopf algebras, and provide a variety of examples. In the case that the Hopf algebras in question are commutative, we also determine criteria for them to be isomorphic as $ K $-algebras. By applying our results, we complete a detailed analysis of the distinct Hopf algebras and $ K $-algebras that appear in the classification of Hopf-Galois structures on a cyclic extension of degree $ p^{n} $, for $ p $ an odd prime number.

math.NT↗

arXiv2017

Normality and Short Exact Sequences of Hopf-Galois Structures

Every Hopf-Galois structure on a finite Galois extension $K/k$ where $G=Gal(K/k)$ corresponds uniquely to a regular subgroup $N\leq B=\operatorname{Perm}(G)$, normalized by $λ(G)\leq B$, in accordance with a theorem of Greither and Pareigis. The resulting Hopf algebra which acts on $K/k$ is $H_N=(K[N])^{λ(G)}$. For a given such $N$ we consider the Hopf-Galois structure arising from a subgroup $P\triangleleft N$ that is also normalized by $λ(G)$. This subgroup gives rise to a Hopf sub-algebra $H_P\subseteq H_N$ with fixed field $F=K^{H_P}$. By the work of Chase and Sweedler, this yields a Hopf-Galois structure on the extension $K/F$ where the action arises by base changing $H_P$ to $F\otimes_k H_P$ which is an $F$-Hopf algebra. We examine this analogy with classical Galois theory, and also examine how the Hopf-Galois structure on $K/F$ relates to that on $K/k$. We will also pay particular attention to how the Greither-Pareigis enumeration/construction of those $H_P$ acting on $K/F$ relates to that of the $H_N$ which act on $K/k$. In the process we also examine short exact sequences of the Hopf algebras which act, whose exactness is directly tied to the descent theoretic description of these algebras.

math.NT↗