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1 Bulletin of the Polish Academy of Sciences. Mathematics

1 2003

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Polynomial imaginary decompositions for finite extensions of fields of characteristic zero

Nowicki A., Spodzieja S.

Bulletin of the Polish Academy of Sciences. Mathematics

2003

Vol. 51, no 2

157-168

Let k be a field of characteristic zero, L = k[xi] a finite field extension of k of degree m > 1. If f is a polynomial in one variable over L, then there exist unique polynomials u0,..., um-1 belonging to k[x0,..., xm-l] such that f(x0 + xix1 + ...xi^m-1 xm-l) = uO + xiu1 + ...xi^m-1 um-1. We prove that for u0, ..., um-1is an element of k[xo,..., xm-1) there exists f for which the above holds if and only if u0, ..., um-1satisfy some generalization of the Cauchy-Riemann equations. Moreover, we show that if f is not an element of L, then the polynomials u0, ... ,um-1 are algebraically independent over k and they have no common divisors in k[xo,... ,Xm-1) of positive degree. Some other properties of polynomials u0,..., um-1 are also given.