# Lindemann–Weierstrass theorem

Lindemann–Weierstrass theorem Part of a series of articles on the mathematical constant π 3.1415926535897932384626433... Uses Area of a circleCircumferenceUse in other formulae Properties IrrationalityTranscendence Value Less than 22/7ApproximationsMemorization People ArchimedesLiu HuiZu ChongzhiAryabhataMadhavaJamshīd al-KāshīLudolph van CeulenFrançois VièteSeki TakakazuTakebe KenkoWilliam JonesJohn MachinWilliam ShanksSrinivasa RamanujanJohn WrenchChudnovsky brothersYasumasa Kanada History ChronologyA History of Pi In culture LegislationPi Day Related topics Squaring the circleBasel problemSix nines in πOther topics related to π vte Part of a series of articles on the mathematical constant e Properties Natural logarithmExponential function Applications compound interestEuler's identityEuler's formulahalf-lives exponential growth and decay Defining e proof that e is irrationalrepresentations of eLindemann–Weierstrass theorem People John NapierLeonhard Euler Related topics Schanuel's conjecture vte In transcendental number theory, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states the following: Lindemann–Weierstrass theorem — if α1, ..., αn are algebraic numbers that are linearly independent over the rational numbers {estilo de exibição mathbb {Q} } , then eα1, ..., eαn are algebraically independent over {estilo de exibição mathbb {Q} } .

Em outras palavras, the extension field {estilo de exibição mathbb {Q} (e^{alfa _{1}},pontos ,e^{alfa _{n}})} has transcendence degree n over {estilo de exibição mathbb {Q} } .

An equivalent formulation (Baker 1990, Capítulo 1, Teorema 1.4), é o seguinte: An equivalent formulation — If α1, ..., αn are distinct algebraic numbers, then the exponentials eα1, ..., eαn are linearly independent over the algebraic numbers.

This equivalence transforms a linear relation over the algebraic numbers into an algebraic relation over {estilo de exibição mathbb {Q} } by using the fact that a symmetric polynomial whose arguments are all conjugates of one another gives a rational number.

The theorem is named for Ferdinand von Lindemann and Karl Weierstrass. Lindemann proved in 1882 that eα is transcendental for every non-zero algebraic number α, thereby establishing that π is transcendental (Veja abaixo).[1] Weierstrass proved the above more general statement in 1885.[2] O teorema, along with the Gelfond–Schneider theorem, is extended by Baker's theorem, and all of these would be further generalized by Schanuel's conjecture.

Conteúdo 1 Naming convention 2 Transcendence of e and π 3 p-adic conjecture 4 Modular conjecture 5 Lindemann–Weierstrass theorem 5.1 Prova 5.1.1 Preliminary lemmas 5.1.2 Final step 5.1.3 Equivalence of the two statements 6 Veja também 7 Notas 8 Referências 9 Leitura adicional 10 External links Naming convention The theorem is also known variously as the Hermite–Lindemann theorem and the Hermite–Lindemann–Weierstrass theorem. Charles Hermite first proved the simpler theorem where the αi exponents are required to be rational integers and linear independence is only assured over the rational integers,[3][4] a result sometimes referred to as Hermite's theorem.[5] Although apparently a rather special case of the above theorem, the general result can be reduced to this simpler case. Lindemann was the first to allow algebraic numbers into Hermite's work in 1882.[1] Shortly afterwards Weierstrass obtained the full result,[2] and further simplifications have been made by several mathematicians, most notably by David Hilbert[6] and Paul Gordan.[7] Transcendence of e and π The transcendence of e and π are direct corollaries of this theorem.

Suppose α is a non-zero algebraic number; então {uma} is a linearly independent set over the rationals, and therefore by the first formulation of the theorem {} is an algebraically independent set; or in other words eα is transcendental. Em particular, e1 = e is transcendental. (A more elementary proof that e is transcendental is outlined in the article on transcendental numbers.) alternativamente, by the second formulation of the theorem, if α is a non-zero algebraic number, então {0, uma} is a set of distinct algebraic numbers, and so the set {e0, } = {1, } is linearly independent over the algebraic numbers and in particular eα cannot be algebraic and so it is transcendental.

To prove that π is transcendental, we prove that it is not algebraic. If π were algebraic, πi would be algebraic as well, and then by the Lindemann–Weierstrass theorem eπi = −1 (see Euler's identity) would be transcendental, uma contradição. Therefore π is not algebraic, which means that it is transcendental.

A slight variant on the same proof will show that if α is a non-zero algebraic number then sin(uma), porque(uma), tan(uma) and their hyperbolic counterparts are also transcendental.