The fundamental theorem of algebra for Hamilton and Cayley numbers

Abstract

In this paper we prove the fundamental theorem of algebra for polynomials with coefficients in the skew field of Hamilton numbers (quaternions) and in the division algebra of Cayley numbers (octonions). The proof, inspired by recent definitions and results on regular functions of a quaternionic and of a octonionic variable, follows the guidelines of the classical topological argument due to Gauss.

On the Multiplicity of Zeroes of Polynomials with Quaternionic Coefficients

“…In what follows, we will simply say polynomials when referring to regular polynomials. Subsequent papers, [1], [4], deepened our understanding of the structure of such polynomials. To begin with, we recall that the product of two regular functions is not regular in general.…”

“…More specifically, given a polynomial and its roots, we show how to construct a factorization in monomials; conversely we will also show how to find the roots of a polynomial if we have one of its factorizations (note that in the complex case, this is clearly a trivial problem: not so in the quaternionic case). These problems were only partially discussed in [1] and in [4]. In this paper, we provide an explicit algorithm that produces the required factorization.…”

“…The first part of the theorem, namely the decomposition (2.1), is Theorem 3.4 in [4]. Thus we only have to prove the decomposition for the quaternionic polynomial Q which has no spherical roots.…”

“…We can assume Q to be a monic polynomial. If not, there is a constant c, coefficient of the highest degree term, and the process below must be preceded by multiplication by c −1 and then followed by multiplication by c. By the fundamental theorem of algebra for quaternionic polynomials, see [4], if deg(Q) = n > 0, there is at least one root, say γ 1 . Let us add the root γ 1 to the polynomial Q(q); this can be accomplished by a simple multiplication and we can now consider the new polynomial…”

A New Approach to Slice Regularity on Real Algebras

“…Jou 1950, Eilenberg-Steenrod 1952, Pogorui-Shapiro 2004, Gentili-Struppa-Vlacci 2008, Ghiloni-P. 2009 Examples 1 In R n every polynomial i x i a i , (a i paravectors) has roots in Q A .…”