We introduce the notions of alternating roots of polynomials and alternating polynomials over a Cayley-Dickson algebra, and prove a connection between the alternating roots of a given polynomial and the roots of the corresponding alternating polynomial over the Cayley-Dickson doubling of the algebra. We also include a detailed Octave code for the computation of alternating roots over Hamilton's quaternions.

• 11R52 - Quaternion and other division algebras: arithmetic, zeta functions
• 17A20 - Flexible algebras
• 17A45 - Quadratic algebras (but not quadratic Jordan algebras)
• 17A75 - Composition algebras
• 17D05 - Alternative rings