Lindemann–Weierstrass theorem

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In mathematics, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states that if α₁,...,α_n are algebraic numbers which are linearly independent over the rational numbers Q, then are algebraically independent over Q; in other words the extension field has transcendence degree n over .

An equivalent formulation (Baxter 1975, Chapter 1, Theorem 1.4), is the following: If α₁,...,α_n are distinct algebraic numbers, then the exponentials are linearly independent over the algebraic numbers.

The theorem is named for Ferdinand von Lindemann and Karl Weierstrass. Lindemann proved in 1882 that e^α is transcendental for every non-zero algebraic number α, thereby establishing that π is transcendental (see below). Weierstrass proved the above more general statement in 1885.

The theorem, along with the Gelfond-Schneider theorem, is generalized by Schanuel's conjecture.

Contents 1 Naming convention 2 Transcendence of e and π 3 p-adic conjecture 4 See also 5 External links 6 References

[edit] 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 α_i are required to be rational integers and linear independence is only assured over the rational integers^[1], a result sometimes referred to as Hermite's theorem^[2]. 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^[3]. Shortly after Weierstrass obtained the full result^[4], and further simplifications have been made by several mathematicians, most notably by David Hilbert.

[edit] Transcendence of e and π

Part of a series of articles on The mathematical constant, e Natural logarithm Applications in: compound interest · Euler's identity & Euler's formula  · half-lives & exponential growth/decay Defining e: proof that e is irrational  · representations of e · Lindemann–Weierstrass theorem People John Napier  · Leonhard Euler Schanuel's conjecture

The transcendence of e and π are direct corollaries of this theorem.

Suppose α is a nonzero algebraic number; then {α} is a linearly independent set over the rationals, and therefore by the first formulation of the theorem {e^α} is an algebraically independent set; or in other words e^α is transcendental. In particular, e¹ = e is transcendental. (A more elementary proof that e is transcendental is outlined in the article on transcendental numbers.)

Alternatively, using the second formulation of the theorem, we can argue that if α is a nonzero algebraic number, then {0, α} is a set of distinct algebraic numbers, and so the set {e⁰,e^α} = {1,e^α} is linearly independent over the algebraic numbers and in particular e^α can't be algebraic and so it is transcendental.

Now, we prove that π is transcendental. If π were algebraic, 2πi would be algebraic too (since 2i is algebraic), and then by the Lindemann-Weierstrass theorem e^2πi = 1 (see Euler's formula) would be transcendental, which is absurd.

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

[edit] p-adic conjecture

The p-adic Lindemann–Weierstrass conjecture is that a p-adic analog of this statement is also true: suppose p is some prime number and α₁,...,α_n are p-adic numbers which are algebraic over Q and linearly independent over Q, such that | α_i | _p < 1 / p for all i; then the p-adic exponentials are p-adic numbers that are algebraically independent over Q.

[edit] See also

• Proof that e is irrational
• Proof that π is irrational

[edit] External links

• (French) Proof's Lindemann-Weierstrass (HTML)

[edit] References

• Baker, Alan (1975), Transcendental Number Theory, Cambridge University Press, ISBN 052139791X 

1. ^ Sur la fonction exponentielle, Comptes Rendus Acad. Sci. Paris, 77, pages 18-24, 1873.
2. ^ A.O.Gelfond, Transcendental and Algebraic Numbers, translated by Leo F. Boron, Dover Publications, 1960.
3. ^ Über die Ludolph'sche Zahl, Sitzungber. Königl. Preuss. Akad. Wissensch. zu Berlin, 2, pages 679-682, 1888.
4. ^ Zu Hrn. Lindemann's Abhandlung: 'Über die Ludolph'sche Zahl' , Sitzungber. Königl. Preuss. Akad. Wissensch. zu Berlin, 2, pages 1067-1086, 1885