Hurwitz’ Theorem

In this article we describe several results based on the paper [Hur98] and which we will refer to as Hurwitz’ theorem. There are several related results: the classification of real normed division algebras, the classification of complex composition algebras and the classification of real composition algebras. The classification of real division algebras is an open problem. Our interest is in composition algebras. This article was originally conceived of as an account of the result of Rost in [Ros96] and the intention was to promote the use of diagrams. This result can also be found in [Cvi08, Chapter 16] although it is less explicit. This is still the core of this article. However I then added a prequel giving some background on composition algebras and vector product algebras. In particular I give brief accounts of two other proofs of Hurwitz’ theorem. Both proofs can be found with more detail in [Bae02, §2]. I then added a sequel based on [DEF99, Volume 1]. The relevant Chapters are Chapter 6 in Notes on Spinors and Chapters 1 & 2 in Supersolutions. The aim here is to explain an algebraic property of the Poincare supergroup and Minkowski superspace and to show that this is satisfied only for Minkowski space of dimension 3, 4, 6 and 10. As a stepping stone to this I have also included a section with an unconventional view on triality. In these notes we work over a field F whose characteristic is not two and for (25) we require that the characterisric is not two or three.