Polynomial imaginary decompositions for finite extensions of fields of characteristic zero

• Autorzy: Nowicki A. , Spodzieja S.
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

polynomial; decomposition; Cauchy-Riemann equations

PL

wielomian; rozkład; równanie Cauchy'ego-Riemanna

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2003
• Tom: Vol. 51, no 2
• Strony: 157--168
• Opis fizyczny: Bibliogr. 7 poz.

Twórcy

autor

Nowicki A.

• Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland

autor

Spodzieja S.

• Faculty of Mathematics, University of Łódź, S. Banacha 22, 90-238 Łódź, Poland

Bibliografia

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• [2] I. J. Good, A simple generalization of analytic function theory, Exposition. Math., 6 (1988) 289-311.
• [3] E. D. Martin, Some elements of a theory of multidimensional complex variables. I. General theory, J. Franklin Inst. B, 326 (1989) 611-647.
• [4] A. Schinzel, Polynomials with special regard to reducibility, Cambridge University Press, Cambridge 2000.
• [5] J. P. Serre, Topics in Galois theory, Jones and Bartlett, Boston 1992.
• [6] W. Sierpiński, Elementary theory of numbers, Hafner Publishing Company, New York 1964.
• [7] N. N. Vorobiev, The Fibonacci numbers (in Russian), Nauka, Moscow 1978.