Factorization of polynomials using commuting matrices
The result in this paper is: Let \(F\) be a field, \(f(x)=p_ 1(x)^{e_ 1} \dots p_ r(x)^{e_ r}\) a monic polynomial of degree \(n\) over \(F\), where \(r \geq 1\), \(p_ 1(x), \dots, p_ r(x)\) are distinct irreducible monic polynomials in \(F[x]\), \(e_ 1,\dots,e_ r\) are positive integers. Then there exist commuting \(n \times n\) matrices \(A_ 1,\dots,A_ n\) with entries in \(F\), each having characteristic and minimal polynomial \(f(x)\), such that \[ f(x)I_ n =(xI_ n-A_ 1)\cdots (xI_ n-A_ n), \] if and only if the following conditions hold: (a) If \(\theta_ i\) is a root of the equation \(p_ i(x)=0\) in a (fixed) algebraic closure of \(F\), then \(F[\theta_ i]\) is the splitting field of \(p_ i(x)\) over \(F\) and \(F[\theta_ 1] =F[\theta_ 2]= \cdots=F [\theta_ r]\). (b) \(e_ 1=e_ 2= \cdots=e_ r=e\), say. (c) \(x^ e-1\) splits into linear factors in \(F[\theta_ 1][x]\). (d) If \(p_ 1(x)\) is not separable over \(F\) (that is, if the formal derivative \(p_ 1'(x)=0\), then \(e\) is a power of the characteristic of the field \(F\). This result is the development of the classical one: if \(F\) is an infinite field, such a factorization without the requirement that \(A_ 1,\dots,A_ n\) commute always exists.
scientific article; zbMATH DE number 519703
scientific article