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