Polynomial Imaginary Decompositions for Finite Separable Extensions

Adam Grygiel

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 56 (2008), p. 9-13 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $u₀,...,u_{m-1} ∈ K[X₀,...,X_{m-1}]$ such that $f(∑_{j=0}^{m-1}ξ^{j}X_{j}) = ∑_{j=0}^{m-1}ξ^{j}u_{j}$. A. Nowicki and S. Spodzieja proved that, if K is a field of characteristic zero and f ≠ 0, then $u₀,...,u_{m-1}$ have no common divisor in $K[X₀,...,X_{m-1}]$ 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.

Détail

Publié le : 2008-01-01

Zbl 1195.12006

EUDML-ID : urn:eudml:doc:281167