Polynomial Imaginary Decompositions for Finite Separable Extensions

Adam Grygiel¹

¹ Faculty of Mathematics and Computer Science University of /L/od/x Banacha 22 90-238 /L/od/x, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 56 (2008) no. 1, pp. 9-13.

Résumé

Let $K$ be a field and let $L=K[\xi]$ be a finite field extension of $K$ of degree $m>1$. If $f\in L[Z]$ is a polynomial, then there exist unique polynomials $u_0,\ldots,u_{m-1}\in K[X_0,\ldots,X_{m-1}]$ such that $f(\sum_{j=0}^{m-1}\xi ^{j}X_{j})=\sum_{j=0}^{m-1}\xi ^{j}u_{j}$. A. Nowicki and S. Spodzieja proved that, if $K$ is a field of characteristic zero and $f\not=0$, then $u_0,\ldots,u_{m-1}$ have no common divisor in $K[X_0,\ldots,X_{m-1}]$ 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

DOI : 10.4064/ba56-1-2

Keywords: field finite field extension degree polynomial there exist unique polynomials ldots m ldots m sum m sum m nowicki spodzieja proved field characteristic zero ldots m have common divisor ldots m positive degree extend result separable extension field arbitrary characteristic formal power series several variables