An extension of a factorization theorem of wedderburn to matrix rings

Abstract

Let F be a field of characteristic zero, and let M n ( F) be the algebra of n × n matrices over F. Let f( x) be a monic polynomial of degree n in F[ x]. It is proved that there exist n × n matrices A 1, …, A n all with minimal polynomial f( x) such that f( x) I n = ( xI n − A 1) ⋯ ( xI n − A n . A simple inductive procedure for constructing A 2, …, A n , having chosen A 1 to be the companion matrix of f( x), is established. The procedure also leads to an improved version of a theorem of Wedderburn on the factorization of certain polynomials over division rings.