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Polynomials with Coefficients from a Division Ring

Published online by Cambridge University Press:  20 November 2018

Una Bray  and
George Whaples
Una Bray
Affiliation:
Smith College, Northampton, Massachusetts
George Whaples
Affiliation:
University of Massachusetts, Amherst, Massachusetts
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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 (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).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 35 , Issue 3 , 01 June 1983 , pp. 509 - 515
Copyright
Copyright © Canadian Mathematical Society 1983

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