A simple and self-contained proof for the Lindemann-Weierstrass theorem

TL;DR

This paper presents an accessible, self-contained proof of the Lindemann-Weierstrass theorem, making advanced algebraic number theory concepts understandable for undergraduates.

Contribution

It offers a simplified, comprehensive proof of the Lindemann-Weierstrass theorem with all necessary algebraic tools fully explained.

Findings

Proof is accessible to undergraduates

All algebraic tools are elementary and fully proved

Clarifies the foundational aspects of the theorem

Abstract

The famous result of Lindemann and Weierstrass says that if $a_{1},a_{2},\ldots,a_{n}$ are distinct algebraic numbers, then $e^{a_{1}},e^{a_{2}},\ldots,e^{a_{n}}$ are linearly independent complex numbers over the field $\overline{\mathbb{Q}}$ of all algebraic numbers. Starting from some basic ideas of Hermite, Lindemann, Hilbert, Hurwitz and Baker, in this note we provide an easy to understand and self-contained proof for the Lindemann-Weierstrass Theorem. In an introductory section we have gathered all the algebraic number theory tools that are necessary to prove the main theorem. All these auxiliary results are fully proved in a simple and elementary way, so that the paper can be read even by an undergraduate student.