Let K/F and K/L be two cyclic Galois field extensions and D = (K/F, σ, c) a cyclic algebra. Given an invertible element d ∈ D, we present three families of unital nonassociative algebras over L ∩ F defined on the direct sum of n copies of D. Two of these families appear either explicitly or implicitly in the designs of fast-decodable space-time block codes in papers by Srinath, Rajan, Markin, Oggier, and the authors.We present conditions for the algebras to be division and propose a construction for fully diverse fast decodable space-time block codes of rate-m for nm transmit and m receive antennas. We present a DMT-optimal rate-3 code for 6 transmit and 3 receive antennas which is fast-decodable, with ML-decoding complexity at most O(M 15 ).1991 Mathematics Subject Classification. Primary: 17A35, 94B05.

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.

Associative division algebras are a rich source of fully diverse space-time block codes (STBCs). In this paper the systematic construction of fully diverse STBCs from nonassociative algebras is discussed. As examples, families of fully diverse $2\times 2$, $2\times 4$ multiblock and $4\x 4$ STBCs are designed, employing nonassociative quaternion division algebras.Comment: 23 pages; final version; to appear in Advances in Mathematics of Communication

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.