An Accessible Proof of Hurwitz’s Sums of Squares Theorem

Abstract

Summary We give a simple proof, intelligible to undergraduates, that a particular multiplicative formula for sums of n squares can only occur when or 8, a result originally proved by Hurwitz in 1898. We begin with a brief survey of the history of sums of squares, leading to a discussion of the related topic of normed division algebras over the real numbers. This story culminates with a crucial paper by Dickson in 1919 that not only contained an exposition of Hurwitz’s 1898 proof, but which also outlined a new process for producing division algebras over the reals. That process, now called the Cayley-Dickson construction, is intimately connected with the product formula for sums of squares and the dimensions necessary for its existence. For this reason, we present an introduction to the Cayley-Dickson construction for beginners, together with a proof of Hurwitz’s theorem accessible to anyone with a basic knowledge of undergraduate algebra.