When is ℝ 2 a Division Algebra?

Abstract

The equivalence of conditions (iii) and (iii') follows from the invertibility of left and right translation (both linear transformations) by nonzero elements of the algebra. Are there other ways of defining multiplication to make R2 a division algebra? There are, of course, division algebras of higher dimension, for example, the quaternions and Cayley numbers. It is well known that multiplication in these systems is completely determined by a multiplication table for a basis and that both are noncommutative, while the Cayley numbers are nonassociative. These facts suggest two things about an investigation of different multiplications for R 2. First, it suffices to study basis multiplication since the multiplicative structure of an algebra is completely determined by a multiplication table for any basis. The proof of this statement involves a straightforward application of properties (i) and (ii) above. Second, the investigation should include algebras that are noncommutative, nonassociative, or even lack a unit. Indeed, the following theorem shows that without easing conditions on the multiplication the search ends abruptly.