Let R be any division ring and let
1
be 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 (c
1, c
2, …, cn
) be a set of pairwise non-conjugate elements of R, and (m
1, …, mN
) positive integers such that Σmi = n = deg f(x).