Potências transcendentes : uma jornada pelo Teorema de Gelfond-Schneider

Abstract

This monograph focuses on the study of the Gelfond-Schneider Theorem, one of the most remarkable results in the theory of transcendental numbers, whose proof solved Hilbert’s seventh problem. The text guides the reader through a gradual journey of fundamental concepts, ranging from the notions of algebraic and transcendental numbers to classical results, such as Liouville’s Theorem, the transcendence of notable constants (such as e and π), and the Hermite-Lindemann Theorem. Then, it presents the methods and auxiliary results that underpin the proof by Gelfond and Schneider, also exploring consequences and applications of the theorem, such as the characterization of the transcendence of various algebraic powers. Thus, this work aims not only to present the proof of this result but also to contextualize its importance for the consolidation of transcendence theory and for solving open problems in modern mathematics.