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 $u₀,...,u_{m-1}\in K[X₀,...,X_{m-1}]$ such that $f\left({\sum }_{j=0}^{m-1}{\xi }^j{X}_j\right)={\sum }_{j=0}^{m-1}{\xi }^j{u}_j$. A. Nowicki and S. Spodzieja proved that, if K is a field of characteristic zero and f ≠ 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....

The fields defined by the polynomials constructed in E. Nart and the author in J. Number Theory 16, (1983), 6–13, Th. 2.1, with absolute Galois group the alternating group $A_n$, can be embedded in any central extension of $A_n$ if and only if $n\equiv 0\left(mod8\right)$, or $n\equiv 2\left(mod8\right)$ and $n$ is a sum of two squares. Consequently, for theses values of $n$, every central extension of $A_n$ occurs as a Galois group over $\mathbf{Q}$.

We give an effective characterization theorem for integral monic irreducible polynomials $f$ of degree $n$ whose Galois groups over $\mathbb{Q}$ are Frobenius groups with kernel of order $n$ and complement of prime order.