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Algebraic Numbers as Product of Powers of Transcendental Numbers

Department of Mathematics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Czech Republic
Symmetry 2019, 11(7), 887; https://doi.org/10.3390/sym11070887
Received: 17 June 2019 / Revised: 3 July 2019 / Accepted: 4 July 2019 / Published: 8 July 2019
(This article belongs to the Special Issue Number Theory and Symmetry)
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Abstract

The elementary symmetric functions play a crucial role in the study of zeros of non-zero polynomials in C [ x ] , and the problem of finding zeros in Q [ x ] leads to the definition of algebraic and transcendental numbers. Recently, Marques studied the set of algebraic numbers in the form P ( T ) Q ( T ) . In this paper, we generalize this result by showing the existence of algebraic numbers which can be written in the form P 1 ( T ) Q 1 ( T ) P n ( T ) Q n ( T ) for some transcendental number T, where P 1 , , P n , Q 1 , , Q n are prescribed, non-constant polynomials in Q [ x ] (under weak conditions). More generally, our result generalizes results on the arithmetic nature of z w when z and w are transcendental. View Full-Text
Keywords: Baker’s theorem; Gel’fond–Schneider theorem; algebraic number; transcendental number Baker’s theorem; Gel’fond–Schneider theorem; algebraic number; transcendental number
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Trojovský, P. Algebraic Numbers as Product of Powers of Transcendental Numbers. Symmetry 2019, 11, 887.

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