Polynomials and degrees of maps in real normed algebras

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