Polynomial in Non-Commutative Algebra

TL;DR

This paper explores the properties of polynomials in non-commutative algebra, including root sets and division, highlighting unique behaviors not seen in commutative cases.

Contribution

It introduces the definition and properties of polynomials in non-commutative algebra, including examples of root sets and polynomial division with remainder.

Findings

Some polynomials have no roots in non-commutative algebra.

Certain polynomials have infinite root sets.

Division of polynomials with remainder is considered.

Abstract

I considered definition and properties of polynomial in no-commutative algebra. There exists polynomial which has finite, infinite or empty set of roots. For instance, the polynomial $$p_1(x)=ix-xi-1$$ have no root and the polynomial $$p_k(x)=ix-xi-k$$ has the set of roots $$x=C_1+C_2i+\frac 12 j$$ I considered division of polynomials with remainder.