Lindemann–Weierstrass theorem

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 $e^\{alpha\_1\},ldots,e^\{alpha\_n\}$ are algebraically independent over Q; in other words the extension field $mathbb\{Q\}(e^\{alpha\_1\},\; ldots,e^\{alpha\_n\})$ has transcendence degree "n" over $mathbb\{Q\}$.

An equivalent formulation Harv|Baxter|1975|loc=Chapter 1, Theorem 1.4, is the following: If α₁,...,α_"n" are distinct algebraic numbers, then the exponentials $e^\{alpha\_1\},ldots,e^\{alpha\_n\}$ 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.

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 ["Sur la fonction exponentielle", Comptes Rendus Acad. Sci. Paris, 77, pages 18-24, 1873.] , a result sometimes referred to as Hermite's theorem [A.O.Gelfond, "Transcendental and Algebraic Numbers", translated by Leo F. Boron, Dover Publications, 1960.] . 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 ["Über die Ludolph'sche Zahl", Sitzungber. Königl. Preuss. Akad. Wissensch. zu Berlin, 2, pages 679-682, 1888.] . Shortly after Weierstrass obtained the full result ["Zu Hrn. Lindemann's Abhandlung: 'Über die Ludolph'sche Zahl' ", Sitzungber. Königl. Preuss. Akad. Wissensch. zu Berlin, 2, pages 1067-1086, 1885] , and further simplifications have been made by several mathematicians, most notably by David Hilbert.

Transcendence of "e" and π

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, "e"^{1 = "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 $\{e^0,e^alpha\}=\{1,e^alpha\}$ 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, 2π"i" would be algebraic too (since 2"i" is algebraic), and then by the Lindemann-Weierstrass theorem "e"^2π"i" = 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.

"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 for all "i"; then the p-adic exponentials $e^\{alpha\_1\},\; ldots,\; e^\{alpha\_n\}$ are "p"-adic numbers that are algebraically independent over Q.

ee also

*Proof that e is irrational
*Proof that π is irrational

Notes

References

*Citation|last=Baker|first=Alan|title=Transcendental Number Theory|publisher=Cambridge University Press|year=1975|isbn=052139791X

External links

* [http://nombrejador.free.fr/article/lindemann-weierstrass_ttj.htm Proof's Lindemann-Weierstrass (HTML)]