  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.
