Algebraic Numbers as Product of Powers of Transcendental Numbers

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.