Polynomial Imaginary Decompositions for Finite Separable Extensions

Abstract

Let K be a field and let L = K[ξ] be a finite field extension of K of degree m > 1. If f ∈ L[Z] is a polynomial, then there exist unique polynomials u0, . . . , um−1 ∈ K[X0, . . . , Xm−1] such that f( Pm−1 j=0 ξ Xj) = Pm−1 j=0 ξ uj . A. Nowicki and S. Spodzieja proved that, if K is a field of characteristic zero and f 6= 0, then u0, . . . , um−1 have no common divisor in K[X0, . . . , Xm−1] of positive degree. We extend this result to the case when L is a separable extension of a field K of arbitrary characteristic. We also show that the same is true for a formal power series in several variables.