Open Access
On Hopf Galois structures and complete groups
TLDR
In this paper, the number of Hopf Galois structures on a Galois extension of K, fields, with Galois group Γ was investigated. But the results were restricted to the case where the associated group of the Hopf algebra H is a safeprime.Abstract:
Let L be a Galois extension of K, fields, with Galois group Γ. We obtain two results. First, if Γ = Hol(Zpe ), we determine the number of Hopf Galois structures on L/K where the associated group of the Hopf algebra H is Γ (i.e. L ⊗K H ∼ L(Γ)). Now let p be a safeprime, that is, p is a prime such that q =( p − 1)/2 > 2 is also prime. If L/K is Galois with group Γ = Hol(Zp), p a safeprime, then for every group G of cardinality p(p−1) there is an H-Hopf Galois structure on L/K where the associated group of H is G, and we count the structures.read more
Citations
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Journal ArticleDOI
Hopf–Galois structures on Galois field extensions of degree pq
TL;DR: In this article, all Hopf-Galois structures on a Galois extension of fields of degree pq were determined, where p, q are primes with p≡1 ( mod q).
Journal ArticleDOI
Abelian Hopf Galois structures on prime-power Galois field extensions
TL;DR: In this paper, it was shown that every regular abelian subgroup of the holomorph of N is isomorphic to (N, +) under the assumption that n is a Hopf Galois extension of fields.
Journal ArticleDOI
Fixed-point free endomorphisms and Hopf Galois structures
TL;DR: In this article, it was shown that fixed point free endomorphisms of a Galois group G yield Hopf Galois structures on L|K. The Hopf structures that arise are twisted versions of the Hopf structure by Hλ, the K-Hopf algebra that arises from the left regular representation of G in Perm(G) normalized by G.
Fixed-point free pairs of homomorphisms and nonabelian Hopf-Galois structure
Nigel P. Byott,Lindsay N. Childs +1 more
TL;DR: In this paper, the authors consider regular embeddings that arise from fixed-point free pairs of homomorphisms from an embedding to a holomorph of a Galois group and compare the results with a count of Hopf-Galois structures obtained by T. Kohl.
Journal ArticleDOI
Elementary abelian Hopf Galois structures and polynomial formal groups
TL;DR: In this paper, the authors find a relationship between regular embeddings of G, an elementary abelian p-group of order p n, into unipotent upper triangular matrices with entries in F p and commutative dimension n degree 2 polynomial formal groups with nilpotent upper triangular structure matrices.
References
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Journal ArticleDOI
Hopf Galois theory for separable field extensions
Cornelius Greither,Bodo Pareigis +1 more
Book
Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory
TL;DR: Hopf algebras and Galois extensions of separable field extensions have been studied in this article, where they have been applied to principal homogeneous spaces and formal groups.
Book
A concrete introduction to higher algebra
TL;DR: In this paper, the authors present an introduction to higher algebra for students with a background of a year of calculus, aiming to give these students enough experience in the algebraic theory of the integers and polynomials to appreciate the basic concepts of abstract algebra.
Journal ArticleDOI
On the hopf galois theory for separable field extensions
TL;DR: In this article, the hopf galois theory for separable field extensions is studied and the authors propose a hopf-galois-based hopf theory for field extensions.