Do teorema de Liouville ao sétimo problema de Hilbert e algumas consequências

Abstract

In this work, we study the development of the theory of algebraic and transcendent numbers with emphasis on a solution of Hilbert’s Seventh Problem, a result that

brought together the efforts of great mathematicians. For a better understanding of

this process, we present the result obtained by Liouville from a theorem that characterizes algebraics, then we build a number that does not satisfy this characterization,

therefore, it will be transcendent. We will prove the remarkable existence of transcendents via Liouville and through Cantor, showing that the infinite of the transcendent

is not enumerable, while of the algebraic it is enumerable, showing that there are many more transcendent numbers than algebraic. We will demonstrate a generalization of

the Lindemann Theorem established by Hermite-Lidemann, with more general consequences such as the transcendence of certain numbers and functions: e^α , e, π, log(α), sin(α), cos(α) and tan(α), being α algebraic, and yet, our main object of study, which is a solution to Hilbert’s Seventh Problem and some consequences. Problem that asked if numbers of the form α^β , where α is an algebraic number different from 0 and 1; and β is an algebraic and irrational number, they are all transcendent. In this sense, we have an infinity of numbers in the form 2^(√2), i^i, log_10(2), e^π e (log 3)/(log 2) that are transcendent.

Finally, as a consequence, we will introduce a recent significant advance of a more general formulation of a conjecture proved by Baker, which says that any finite non-zero

combination of algebraic logarithms with algebraic coefficients is transcendent, and thus, facilitating the search for transcendents and enabling the development of other areas.