# Lindemann-Weierstrass theorem

If $\alpha_{1},\ldots,\alpha_{n}$ are linearly independent algebraic numbers over $\mathbb{Q}$, then $e^{\alpha_{1}},\ldots,e^{\alpha_{n}}$ are algebraically independent over $\mathbb{Q}$.

An equivalent version of the theorem that if $\alpha_{1},\ldots,\alpha_{n}$ are distinct algebraic numbers over $\mathbb{Q}$, then $e^{\alpha_{1}},\ldots,e^{\alpha_{n}}$ are linearly independent over $\mathbb{Q}$.

Some immediate consequences of this theorem:

• If $\alpha$ is a non-zero algebraic number over $\mathbb{Q}$, then $e^{\alpha}$ is transcendental over $\mathbb{Q}$.

• $e$ is transcendental over $\mathbb{Q}$.

• $\pi$ is transcendental over $\mathbb{Q}$. As a result, it is impossible to “square the circle”!

It is easy to see that $\pi$ is transcendental over $\mathbb{Q}(e)$ iff $e$ is transcendental over $\mathbb{Q}(\pi)$ iff $\pi$ and $e$ are algebraically independent. However, whether $\pi$ and $e$ are algebraically independent is still an open question today.

Schanuel’s conjecture is a generalization of the Lindemann-Weierstrass theorem. If Schanuel’s conjecture were proven to be true, then the algebraic independence of $e$ and $\pi$ over $\mathbb{Q}$ can be shown.

 Title Lindemann-Weierstrass theorem Canonical name LindemannWeierstrassTheorem Date of creation 2013-03-22 14:19:22 Last modified on 2013-03-22 14:19:22 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 11 Author CWoo (3771) Entry type Theorem Classification msc 12D99 Classification msc 11J85 Synonym Lindemann’s theorem Related topic SchanuelsConjecutre Related topic GelfondsTheorem Related topic Irrational Related topic EIsTranscendental