An Accessible Proof of Hurwitz’s Sums of Squares Theorem

Summary

We give a simple proof, intelligible to undergraduates, that a particular multiplicative formula for sums of n squares can only occur when $n=1,2,4,$ 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.

MSC::

• Primary 11E25
• Secondary 11R52

Notes

* A more detailed discussion is contained in a recent paper by the authors [11].

* We explain precisely why Hamilton was unable to come up with a three-dimensional extension of $\mathbb{C}$ in a different article [12].

* Dickson’s idea had first appeared in a paper on linear algebras in 1912 [5, pp. 72–73] and was explained in more detail in a book on the subject in 1914 [6, pp. 15–16].

* The equivalence of formulae (2) and (9) is not coincidental!

* Given any distinct $B_i,B_j,B_k$, the fact that $B_i^2=B_j^2=B_k^2=B_i{B}_j{B}_k=-I_n$ is not a coincidence either!

* Notice that this new relation $I_n=\sum c_i{T}_i$ is also irreducible since we can multiply it through by a suitable constant c_i and matrix B_j to return us to our original irreducible R = 0.

Additional information

Notes on contributors

Ezra Brown

Ezra (Bud) Brown (MR Author ID: 222489) grew up in New Orleans, has degrees from Rice and LSU, taught at Virginia Tech for 48 years, and retired in 2017 as Alumni Distinguished Professor Emeritus of Mathematics. He has done research in number theory, combinatorics, and expository mathematics—but one of his favorite papers is one he wrote with a sociologist. He and the late Richard Guy are the authors of the Carus Monograph, The Unity of Combinatorics, published by the AMS in May 2020.

Adrian Rice

Adrian Rice (MR Author ID: 601492) is the Dorothy and Muscoe Garnett Professor of Mathematics at Randolph-Macon College in Ashland, Virginia, where his research focuses on nineteenth-century and early twentieth-century mathematics. In addition to papers on various aspects of the history of mathematics, his books include Mathematics Unbound: The Evolution of an International Mathematical Research Community, 1800–1945 (with Karen Hunger Parshall), Mathematics in Victorian Britain (with Raymond Flood and Robin Wilson), and most recently Ada Lovelace: The Making of a Computer Scientist (with Christopher Hollings and Ursula Martin). He is a five-time recipient of awards for outstanding expository writing from the MAA. In his spare time, he enjoys music, travel, and spending time with his wife and son.