Polynomial Imaginary Decompositions for Finite Separable Extensions

Adam Grygiel. "Polynomial Imaginary Decompositions for Finite Separable Extensions." Bulletin of the Polish Academy of Sciences. Mathematics 56.1 (2008): 9-13. <http://eudml.org/doc/281167>.

@article{AdamGrygiel2008,
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 $u₀,...,u_\{m-1\} ∈ K[X₀,...,X_\{m-1\}]$ 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.},
author = {Adam Grygiel},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {polynomial; decomposition; separable extension},
language = {eng},
number = {1},
pages = {9-13},
title = {Polynomial Imaginary Decompositions for Finite Separable Extensions},
url = {http://eudml.org/doc/281167},
volume = {56},
year = {2008},
}

TY - JOUR
AU - Adam Grygiel
TI - Polynomial Imaginary Decompositions for Finite Separable Extensions
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2008
VL - 56
IS - 1
SP - 9
EP - 13
AB - 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.
LA - eng
KW - polynomial; decomposition; separable extension
UR - http://eudml.org/doc/281167
ER -