On division algebras

Abstract

? 1. The object of this paper is to develop some of the simpler properties of division algebras, that is to say, linear associative algebras in which division is possible by any element except zero. The determination of all such algebras in a given field is one of the most interestiiig problems in the theory of linear algebras. Early in the development of the subject, Frobenius showed that quaternions and its subalgebras form the only division algebras in the field of real numbers and, with the exception of the single theorem that there is no non-commutative division algebra in a filnite field, no further definite result of importance was known till Dickson discovered the algebra referred to in ? 4. It is shown in the present paper that the Dickson algebra is the only noncommutative algebra of order 9 so that the only division algebras of order not greater than 9 are (i) the Dickson algebras of order 4 and 9, (ii) the ordinarv commutative fields, (iii) algebras of order 8 which reduce to a Dickson algebra of order 4 when the field is extended to include those elements of the algebra which are commutative with every other element. ? 2. LEMMA 1. If B is a subalgebra of order b in a division algebra A of order a, there exists a complex C of order c such that