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Polynomial imaginary decompositions for finite separable extensions

Grygiel A.

Bulletin of the Polish Academy of Sciences. Mathematics

2008

Vol. 56, no 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 ƒ ∈ L[Z] is a polynomial, then there exist unique polynomials uo, ..., um-1 ∈ K[Xo, ...,Xm-i] such that ƒ... [wzór]. A. Nowicki and S. Spodzieja proved that, if K is a field of characteristic zero and ƒ ≠ 0, then uo,..., um-1 have no common divisor in K[Xo,..., Xm-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.