Related papers: Hopf Galois Structures on Primitive Purely Insepar…

We study the nonclassical Hopf-Galois module structure of rings of algebraic integers in some extensions of $ p $-adic fields and number fields which are at most tamely ramified. We show that if $ L/K $ is an unramified extension of $ p…

The Hopf-Galois structures on normal extensions $K/k$ with $G=Gal(K/k)$ are in one-to-one correspondence with the set of regular subgroups $N\leq B=Perm(G)$ that are normalized by the left regular representation $\lambda(G)\leq B$. Each…

For an odd prime number $p$, we consider degree $p$ extensions $L/K$ of $p$-adic fields with normal closure $\widetilde{L}$ such that the Galois group of $\widetilde{L}/K$ is the dihedral group of order $2p$. We shall prove a complete…

For a Hopf-Galois structure on a Galois extension $L/K$ of fields that arises from a finite nilpotent $\mathbb{F}_p$-algebra $A$, we look at the Galois correspondence ratio, which measures the failure of surjectivity of the Galois…

Let $H$ be a finite-dimensional Hopf algebra. We study the behaviou r of primitive and maximal ideals in certain types of ring extensions determined by $H$. The main focus is on the class of faithfully flat Galois extensions, which includes…

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 $\lambda(G)\leq B$, in accordance with a theorem of Greither and…

Let $L/K$ be a finite Galois extension of local fields. The Hasse-Arf theorem says that if Gal$(L/K)$ is abelian then the upper ramification breaks of $L/K$ must be integers. We prove the following converse to the Hasse-Arf theorem: Let $G$…

Let $K$ be a local field of characteristic $p>0$ with perfect residue field and let $G$ be a finite $p$-group. In this paper we use Saltman's construction of a generic $G$-extension of rings of characteristic $p$ to construct totally…

Let $K$ be a finite field extension of $\Q$ and let $N$ be a finite group with automorphism group $F=\Aut(N)$. R. Haggenm\"{u}ller and B. Pareigis have shown that there is a bijection \[\Theta: {\mathcal Gal}(K,F)\rightarrow {\mathcal…

Let $A$ be a finite commutative nilpotent $\mathbb{F}_p$-algebra structure on $G$, an elementary abelian group of order $p^n$. If $K/k$ is a Galois extension of fields with Galois group $G$ and $A^p = 0$, then corresponding to $A$ is an…

An extension B\subset A of algebras over a commutative ring k is an H-extension for an L-bialgebroid H if A is an H-comodule algebra and B is the subalgebra of its coinvariants. It is H-Galois if the canonical map A\otimes_B A\to A\otimes_L…

We investigate Hopf-Galois structures on a cyclic field extension $L/K$ of squarefree degree $n$. By a result of Greither and Pareigis, each such Hopf-Galois structure corresponds to a group of order $n$, whose isomorphism class we call the…

Let S/R be a finite extension of discrete valuation rings of characteristic p>0, and suppose that the corresponding extension L/K of fields of fractions is separable and is H-Galois for some K-Hopf algebra H. Let D_{S/R} be the different of…

We provide necessary and sufficient conditions to extend the Hopf-Galois algebra structure on an algebra R to a generalized ambiskew ring based on R, in a way such that the added variables for the extension are skew-primitive in an…

We prove that a depth two Hopf subalgebra K of a semisimple Hopf algebra H is normal (where the ground field $k$ is algebraically closed of characteristic zero). This means on the one hand that a Hopf subalgebra is normal when inducing…

In our previous paper we describe the Galois module structures of $p$th-power class groups $K^\times/{K^{\times p}}$, where $K/F$ is a cyclic extension of degree $p$ over a field $F$ containing a primitive $p$th root of unity. Our…

Let $R$ be a characteristic $p$ discrete valuation ring with field of fractions $K$. Let $H$ be a commutative, cocommutative $K$-Hopf algebra of $p$-power rank which is generated as a $K$-algebra by primitive elements. We construct all of…

Cyclic, ramified extensions $L/K$ of degree $p$ of local fields with residue characteristic $p$ are fairly well understood. Unless $\mbox{char}(K)=0$ and $L=K(\sqrt[p]{\pi_K})$ for some prime element $\pi_K\in K$, they are defined by an…

Given a finite group $ G $, we study certain regular subgroups of the group of permutations of $ G $, which occur in the classification theories of two types of algebraic objects: skew left braces with multiplicative group isomorphic to $ G…

We determine the Hopf Galois structures on a Galois field extension of degree twice an odd prime square and classify the corresponding left braces. Besides we determine the separable field extensions of degree twice an odd prime square…