Algebraic Numbers as Product of Powers of Transcendental Numbers

Pavel Trojovský

doi:10.3390/sym11070887

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.

Highlights

The name “transcendental”, which comes from the Latin word “transcenděre”, was first used for a mathematical concept by Leibniz in 1682
Transcendental numbers in the modern sense were defined by Leonhard Euler
The previous results do not apply, e.g., to prove the existence of algebraic numbers which can be written in the form ( T 2 + 1) T · T T +T +1, with T

Summary

Introduction

The name “transcendental”, which comes from the Latin word “transcenděre”, was first used for a mathematical concept by Leibniz in 1682. Marques [10] showed that the answer for the previous question is yes He proved that for any fixed, non-constant polynomials P( x ), Q( x ) ∈ Q[ x ], the set of algebraic numbers of the form P( T )Q(T ) , with T transcendental, is dense in some connected subset of either R or C. The previous results do not apply, e.g., to prove the existence of algebraic numbers which can be written in the form ( T 2 + 1) T · T T +T +1 , with T transcendental. The proof of the above theorem combines famous classical theorems concerning transcendental numbers (like the Baker’s Theorem on linear forms in logarithms and the Gel’fond–Schneider Theorem) and certain purely field-theoretic results. In a similar way, we can prove Theorem 2 for rational functions with algebraic coefficients, but we choose to prove this simpler case in order to avoid too many technicalities, which can obscure the essence of the main idea

Auxiliary Results

The Proof

Conclusions