Automorphisms and isomorphisms of Jha-Johnson semifields obtained from skew polynomial rings

Abstract

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.

Nonassociative cyclic extensions of fields and central simple algebras

“…Then ker(N K/F ) is a cyclic group of order s = (q m − 1)/(q − 1) and any division algebra (K/F, σ, d) has exactly s inner automorphisms, all of them extending id K . The subgroup they generate is cyclic and isomorphic to ker(N K/F ) [10]. Hence if m divides s, which is the case if F contains a primitive mth root of unity, then there is a subgroup of automorphisms of order m extending id K and hence (K/F, σ, d) is a cyclic extension of K of degree m.…”

New semifields and new MRD codes from skew polynomial rings

“…Theorem 8. All semifields of order q 4 with centre of order q and right nucleus of order q 2 are isotopic to a semifield of the form S n,s,1 (η, ρ), for some η ∈ F q n , ρ ∈ Aut(F q n ), [n, s] ∈ { [2,2], [4,1]}.…”

“…If n = 2, s = 2, k = 1, then we are in the case of Section 4.1. So assume n > 2 and [k, s] = [1,2]. Then g = g 0 + g 1 x + g 2n−1 (x 2n−1 + F 1 x n−1 ).…”

The multiplicative loops of Jha-Johnson semifields

“…Let s = (q m − 1)/(q − 1). The following results are implied by the theorems on the automorphism groups for the related algebras S f [3,Theorems 19,20]:…”

“…K = F [x]/(x 2 − 2) = {0, 1, 2, x, 2x, x+1, x+2, 2x+1, 2x+2}. There are exactly two non-isomorphic semifields which are nonassociative quaternion algebras with nucleus K · 1, given by A 1 = (K/F, σ, x) with Aut F (A 1 ) ∼ = Z/4Z and by A 2 = (K/F, σ, x + 1) with Aut F (A 2 ) isomorphic to the group of quaternion units, the smallest dicyclic group Dic 2 (of order 8) [3,Example 32]. These are up to isomorphism the only two proper semifields of order 81 with center F · 1 and nucleus containing K · 1 [W, Theorem 1].…”