FACTORIZATION OF POLYNOMIALS USING COMMUTING MATRICES

Let F be a field, and let M(n)(F) be the algebra of n X n matrices wit h entries in F. Let f(x) is-an-element-of F[x] be a monic polynomial o f degree n. In this paper we find necessary and sufficient conditions on f(x) for the existence of a factorization f(x)I(n) = (xI(n) - A1 .. . (xI(n) - A(n)), where A1,..., A(n) are commuting elements of M(n)(F) , all with minimal polynomial f(x). In an earlier paper it was shown t hat if F is an infinite field, such a factorization without the requir ement that A1,..., A(n) commute always exists.