Polynomials and degrees of maps in real normed algebras

Let be the algebra of quaternions ℍ or octonions . In this manuscript an elementary proof is given, based on ideas of Cauchy and D’Alembert, of the fact that an ordinary polynomial f(t) ∈ [t] has a root in . As a consequence, the Jacobian determinant |J(f)| is always nonnegative in . Moreover, using the idea of the topological degree we show that a regular polynomial g(t) over has also a root in . Finally, utilizing multiplication (*) in , we prove various results on the topological degree of products of maps. In particular, if S is the unit sphere in and h1, h2 : S → S are smooth maps, it is shown that deg(h1 * h2) = deg(h1) + deg(h2)