Polynomials with Coefficients from a Division Ring

Abstract

Let R be any division ring and let1be a polynomial, in the indeterminate X, with coefficients in R. Note that the powers of X are always to the right of the coefficients. We denote the set of all such polynomials by R[X].B. Beck [3] proved the following theorem for the generalized quaternion division algebra; i.e., any division ring of dimension 4 over its center:THEOREM 1. If f(X) is of degree n then f(X) has either infinitely many or at most n zeros in R.Under a reasonable definition of multiplicity Beck also proved:THEOREM 2. Let (c1, c2, …, cn) be a set of pairwise non-conjugate elements of R, and (m1, …, mN) positive integers such that Σmi = n = deg f(x).