A note on the fundamental theorem of algebra for the octonions (Q2475442)

The paper is devoted to a generalization of the fundamental theorem of algebra to the algebra of Cayley numbers \(\mathbb{K}\). It is proved that any noncommutative nonassociative polynomial with coefficients in \(\mathbb{K}\) containing a unique term of the highest degree has a root in \(\mathbb{K}\). As a corollary the following theorem is obtained: Let \(a\) be a non-real element of an algebra \(A\) obtained by the Cayley-Dickson process over the field of reals. Then the equation \(x^n=a\) has exactly \(n\) solutions in \(A\).