The Hermite–Lindemann–Weierstraß Transcendence Theorem

Manuel Eberl

3 March 2021

Abstract

This article provides a formalisation of the Hermite-Lindemann-Weierstraß Theorem (also known as simply Hermite-Lindemann or Lindemann-Weierstraß). This theorem is one of the crowning achievements of 19th century number theory.

The theorem states that if $\alpha_1, \ldots, \alpha_n\in\mathbb{C}$ are algebraic numbers that are linearly independent over $\mathbb{Z}$, then $e^{\alpha_1},\ldots,e^{\alpha_n}$ are algebraically independent over $\mathbb{Q}$.

Like the previous formalisation in Coq by Bernard, I proceeded by formalising Baker's version of the theorem and proof and then deriving the original one from that. Baker's version states that for any algebraic numbers $\beta_1, \ldots, \beta_n\in\mathbb{C}$ and distinct algebraic numbers $\alpha_i, \ldots, \alpha_n\in\mathbb{C}$, we have $\beta_1 e^{\alpha_1} + \ldots + \beta_n e^{\alpha_n} = 0$ if and only if all the $\beta_i$ are zero.

This has a number of direct corollaries, e.g.:

  • $e$ and $\pi$ are transcendental
  • $e^z$, $\sin z$, $\tan z$, etc. are transcendental for algebraic $z\in\mathbb{C}\setminus\{0\}$
  • $\ln z$ is transcendental for algebraic $z\in\mathbb{C}\setminus\{0, 1\}$
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