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Abstract

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.