Zeros of quaternion polynomials

Abstract

It is well known that, over a division ring, every zero of a polynomial f( x) = ( x − x 1)…( x − x n ) is congruent to x r for some r. In this note, we show further that, over the quaternion field, there exists at least one quaternion q r congruent to each x r , and that, through this result, a constructive method for determining the zeros of quaternion polynomials can be established.