Polynomial Imaginary Decompositions for Finite Separable Extensions

• Autorzy: Adam Grygiel
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 12F10: Separable extensions, Galois theory
• 12E05: Polynomials (irreducibility, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2008
• Tom: 56
• Numer: 1
• Strony: 9-13

Daty

wydano

2008

Twórcy

autor

Adam Grygiel

• Faculty of Mathematics and Computer Science, University of Łódź, Banacha 22, 90-238 Łódź, Poland

Identyfikatory

DOI

10.4064/ba56-1-2