Abstract. Let σ be an automorphism of a field K with fixed field F . We study the automorphisms of nonassociative unital algebras which are canonical generalizations of the associative quotient algebras K[t; σ]/f K[t; σ] obtained when the twisted polynomial f ∈ K[t; σ] is invariant, and were first defined by Petit. We compute all their automorphisms if σ commutes with all automorphisms in Aut F (K) and n ≥ m − 1, where n is the order of σ and m the degree of f , and obtain partial results for n < m − 1. In the case where K/F is a finite Galois field extension, we obtain more detailed information on the structure of the automorphism groups of these nonassociative unital algebras over F . We also briefly investigate when two such algebras are isomorphic.
We study the automorphisms of Jha-Johnson semifields obtained from a right invariant irreducible twisted polynomial f ∈ K[t; σ], where K = F q n is a finite field and σ an automorphism of K of order n, with a particular emphasis on inner automorphisms and the automorphisms of Sandler and Hughes-Kleinfeld semifields. We include the automorphisms of some Knuth semifields (which do not arise from skew polynomial rings).Isomorphism between Jha-Johnson semifields are considered as well.Every finite nonassociative Petit algebra is a Jha-Johnson semifield. These algebras were studied by Wene [35] and more recently, Lavrauw and Sheekey [23].While each Jha-Johnson semifield is isotopic to some such algebra S f it is not necessarily itself isomorphic to an algebra S f . We will focus on those Jha-Johnson semifields which are, and apply the results from [12] to investigate their automorphisms.The structure of the paper is as follows: In Section 1, we introduce the basic terminology and define the algebras S f . Given a finite field K = F q n , an automorphism σ of K of order n with F = Fix(σ) = F q and an irreducible polynomialwe know the automorphisms of the Jha-Johnson semifields S f if n ≥ m − 1 and a subgroup of them if n < m − 1 [12, Theorems 4, 5]. The automorphism groups of Sandler semifields [30] (obtained by choosing n ≥ m and f (t) = t m − a ∈ K[t; σ], a ∈ K \ F ) are particularly relevant: for all Jha-Johnson semifields. We summarize results on the automorphism groups, and give examples when it is trivial and when Aut F (S f ) ∼ = Z/nZ (Theorem 4). Inner automorphisms of Jha-Johnson semifields are considered in Section 2. In Section 3 we consider the special case that n = m and f (t) = t m − a. In this case, the algebras S f are examples of Sandler semifields and also called nonassociative cyclic algebras (K/F, σ, a). The automorphisms of A = (K/F, σ, a) extending id are inner and form a cyclic group isomorphic to ker(N K/F ). We show when Aut F (A) ∼ = ker(N K/F ) and hence consists only of inner automorphisms, when Aut F (A) contains or equals the dicyclic group Dic r of order 4r = 2q + 2, or when Aut F (A) ∼ = Z/(s/m)Z ⋊ q Z/(m 2 )Z contains or equals a semidirect product, where s = (q m − 1)/(q − 1), m > 2 (Theorems 19 and 20). We compute the automorphisms for the Hughes-Kleinfeld and most of the Knuth semifields in Section 4. Not all Knuth semifields are algebras S f , however, the automorphisms behave similarly in all but one case. We compute the automorphism groups in some examples, improving results obtained by Wene [34]. In Section 5 we briefly investigate the isomorphisms between two semifields S f and S g . In particular, we classify nonassociative cyclic algebras of prime degree up to isomorphism.Sections of this work are part of the first and last author's PhD theses [11,33] written under the supervision of the second author.
We define nonassociative cyclic extensions of degree m of both fields and central simple algebras over fields. If a suitable field contains a primitive mth (resp., qth) root of unity, we show that suitable nonassociative generalized cyclic division algebras yield nonassociative cyclic extensions of degree m (resp., qs). Some of Amitsur's classical results on non-commutative associative cyclic extensions of both fields and central simple algebras are obtained as special cases.2010 Mathematics Subject Classification. Primary: 17A35; Secondary: 17A60, 16S36.
Let S be a unital associative division ring and S[t; σ, δ] be a skew polynomial ring, where σ is an endomorphism of S and δ a left σ-derivation. For each f ∈ S[t; σ, δ] of degree m > 1 with a unit as leading coefficient, there exists a unital nonassociative algebra whose behaviour reflects the properties of f . These algebras yield canonical examples of right division algebras when f is irreducible. The structure of their right nucleus depends on the choice of f . In the classical literature, this nucleus appears as the eigenspace of f , and is used to investigate the irreducible factors of f . We give necessary and sufficient criteria for skew polynomials of low degree to be irreducible. These yield examples of new division algebras S f .
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.