Lindemann-Weierstrass theorem

If ${\alpha }_1,\mathrm{\dots },{\alpha }_n$ are linearly independent algebraic numbers over $\mathbb{Q}$, then $e^{{\alpha }_1},\mathrm{\dots },e^{{\alpha }_n}$ are algebraically independent over $\mathbb{Q}$.

An equivalent version of the theorem that if ${\alpha }_1,\mathrm{\dots },{\alpha }_n$ are distinct algebraic numbers over $\mathbb{Q}$, then $e^{{\alpha }_1},\mathrm{\dots },e^{{\alpha }_n}$ 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