Multiplication on Rⁿ: 1. Division algebra and vector product

Abstract

In 1843 Hamilton discovered the quaternions and in the same year Graves found an algebra with 8 basis elements. So, mathematicians knew the division algebra structures (over R) on R, R², R⁴ and R⁸ in the first half of the 19th century itself. It took more than 100 years to prove that these are all the division algebras over R. This follows from a theorem of Adams (which we will discuss in Part 2). In this part we also discuss 'vector products' on R³ and R⁷ and Hurwitz's theorem on 'sums of squares formulae'.