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单词 LindemannWeierstrassTheorem
释义

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.

TitleLindemann-Weierstrass theorem
Canonical nameLindemannWeierstrassTheorem
Date of creation2013-03-22 14:19:22
Last modified on2013-03-22 14:19:22
OwnerCWoo (3771)
Last modified byCWoo (3771)
Numerical id11
AuthorCWoo (3771)
Entry typeTheorem
Classificationmsc 12D99
Classificationmsc 11J85
SynonymLindemann’s theorem
Related topicSchanuelsConjecutre
Related topicGelfondsTheorem
Related topicIrrational
Related topicEIsTranscendental
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更新时间:2025/5/4 17:00:25