The degree of an eight-dimensional real quadratic division algebra is 1, 3, or 5

TL;DR

This paper proves that 8-dimensional real quadratic division algebras can only have degrees 1, 3, or 5, advancing the classification of such algebras and resolving an open problem in the field.

Contribution

The paper establishes that all 8-dimensional real quadratic division algebras have degrees 1, 3, or 5, narrowing the possibilities and contributing to their classification.

Findings

Degrees of 8-dimensional real quadratic division algebras are limited to 1, 3, or 5.

The result is sharp, meaning no other degrees are possible.

Advances the classification of real quadratic division algebras.

Abstract

A celebrated theorem of Hopf, Bott, Milnor, and Kervaire states that every finite-dimensional real division algebra has dimension 1, 2, 4, or 8. While the real division algebras of dimension 1 or 2 and the real quadratic division algebras of dimension 4 have been classified, the problem of classifying all 8-dimensional real quadratic division algebras is still open. We contribute to a solution of that problem by proving that every 8-dimensional real quadratic division algebra has degree 1, 3, or 5. This statement is sharp.