On Division Algebras
Transactions of the American Mathematical Society
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 interesting 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
... ion of the single theorem that there is no non-commutative division algebra in a finite 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 ordinary 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 A = BC, a = 6c. Denoting elements of B by y with appropriate suffixes, let x2 be an element of A which does not lie in B ; the order of the complex B + Bx2 is then 26 as otherwise there would be a relation of the form yi + y2 x2 = 0, ( y2 + 0 ), which would lead to i2 = -yïl yx < B. Similarly, if x3 < B + Bx2, the order of B + Bx2 + 7Jx3 is 36 since otherwise there would be a relation of the form yx + y2 x2 + y3 x3 = 0, iy3 #= 0), which would lead to x3 = -t/i1 yx -yïl yzx2 < B + Bx2.