Lindemann-Weierstrass theorem

If α1,,αn are linearly independentMathworldPlanetmath algebraic numbersMathworldPlanetmath over , then eα1,,eαn are algebraically independentMathworldPlanetmath over .

An equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath version of the theoremMathworldPlanetmath that if α1,,αn are distinct algebraic numbers over , then eα1,,eαn are linearly independent over .

Some immediate consequences of this theorem:

  • If α is a non-zero algebraic number over , then eα is transcendental over .

  • e is transcendental over .

  • π is transcendental over . As a result, it is impossible to “square the circle”!

It is easy to see that π is transcendental over (e) iff e is transcendental over (π) iff π and e are algebraically independent. However, whether π and e are algebraically independent is still an open question today.

Schanuel’s conjecture is a generalizationPlanetmathPlanetmath of the Lindemann-Weierstrass theoremMathworldPlanetmath. If Schanuel’s conjecture were proven to be true, then the algebraic independence of e and π over 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