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Polynomial Imaginary Decompositions for Finite Separable Extensions

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Grygiel A.

Bulletin of the Polish Academy of Sciences. Mathematics

2008

tom 56

nr 1

9-13

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.

Ograniczanie wyników

1 Bulletin of the Polish Academy of Sciences. Mathematics

1 Grygiel A.

1 2008

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Polynomial Imaginary Decompositions for Finite Separable Extensions