On Octonionic Polynomials

Abstract

We discuss the generalization of results on quaternionic polynomials to the octonionic polynomials. In contrast to the quaternions the octonionic multiplication is non-associative. This fact although introducing some difficulties nevertheless leads to some new results. For instance, the monic and non-monic polynomials do not have, in general, the same set of zeros.Concerning the zeros, it is shown that in the monic and non-monic cases they are not the same, in general, but they belong to the same set of conjugacy classes.Despite these difficulties created by the non-associativity, we obtain equivalent results to the quaternionic case with respect to the number of zeros and the procedure to compute them. Mathematics Subject Classification (2000). 11R52, 20G20.

Construction of Octonionic Polynomials

“…They are the so called Moufang identities (see, for example, [6] for proofs): Finally, we present a theorem and a corollary that may be found in [10], that will be useful in the last section where we present some methods to construct octonionic polynomials.…”

“…In this sense, in this section we will give a summary of the main results on octonionic polynomials. For more detail and for proofs see [10].…”

“…This last theorem gave rise to the algorithm to compute the zeros of an octonionic polynomial (see [10]). …”

“…In a previous paper [10] we defined the basis of the octonionic polynomials, developed the theory and obtained an algorithm to calculate all the zeros of an octonionic polynomial. It remained an important question to be answered: how can we construct an octonionic polynomial with a prescribed set of zeros?…”

“…So we only present here the main results needed for this paper. All the proofs can be found in [10].…”