Finite power-associative division rings

Abstract

The classical Wedderburn theorem [5, p. 37] states that any finite associative division ring is a (commutative) field. A. A. Albert generalized this to finite strictly power-associative division rings of characteristic '-2. His proof used the classification of central simple Jordan algebras and proceeded by case-checking (types A, B, C, D in [1, p. 301] and type ? in [2, p. 11]). The purpose of this paper is to give a uniform proof of his results. Throughout the paper all algebras will be nonassociative algebras over a field 4 of characteristic 3 2; since simple rings (in particular, division rings) are simple algebras over their centroids there is no loss in generality in restricting ourselves to algebras. An algebra is a division algebra if left and right multiplications by a nonzero element are bijections; for finite-dimensional algebras this is equivalent to the nonexistence of proper zero divisors. Following N. Jacobson, we define a Jordan division algebra to be a commutative Jordan algebra with identity element such that every nonzero element x is regular with Jordan inverse y: xy = 1, x2y = x. For special algebras the inverse is just the usual inverse in the associative sense. An algebraic Jordan algebra is a Jordan division algebra if and only if each nonzero x generates a subfield 4 [x], the inverse being a polynomial in x [7, p. 1157]. (Note that this condition is weaker than being a division algebra-if e is an associative quaternion division algebra then D+ is a Jordan division algebra with zero divisors). The following lemma is due to Albert [1, p. 299].