Anders Lindberg: Transcendenta tal

Abstract: This thesis deals with transcendental numbers, focusing on their definition, historical development, and the central results that form the foundation of the theory of these numbers. The thesis begins with a presentation of algebraic and transcendental numbers, as well as fundamental concepts in set theory and Diophantine approximation, which are important building blocks for the subsequent discussion. Next, Cantor’s argument on enumerability is examined, followed by Liouville’s construction of the first explicitly known transcendental numbers and Hermite’s and Lindemann’s proofs of the transcendence of e and π, respectively. Furthermore, the Gelfond–Schneider theorem and its significance for the development of the theory are presented. Finally, further problems and open questions in the field, such as transcendence measures and Schanuel’s conjecture, are discussed, illustrating the ongoing nature of research in this area. The aim of the thesis is to provide a comprehensive, yet detailed, introduction to the most important results and ideas concerning transcendental numbers. I would like to express my sincere gratitude to my supervisor, Rikard Bögvad, for his constructive guidance during this work