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Lindemann–Weierstrass theorem

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:

In other words, the extension field has transcendence degree n over .


An equivalent formulation (Baker 1990, Chapter 1, Theorem 1.4), is the following:


This equivalence transforms a linear relation over the algebraic numbers into an algebraic relation over 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 (see below).[1] Weierstrass proved the above more general statement in 1885.[2]


The theorem, along with the Gelfond–Schneider theorem, is extended by Baker's theorem,[3] and all of these would be further generalized by Schanuel's conjecture.

Naming convention[edit]

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,[4][5] a result sometimes referred to as Hermite's theorem.[6] Although that appears to be a 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[7] and Paul Gordan.[8]

Related result[edit]

A variant of Lindemann–Weierstrass theorem in which the algebraic numbers are replaced by the transcendental Liouville numbers (or in general, the U numbers) is also known.[12]

Gelfond–Schneider theorem

; an extension of Gelfond–Schneider theorem

Baker's theorem

; if proven, it would imply both the Gelfond–Schneider theorem and the Lindemann–Weierstrass theorem

Schanuel's conjecture

(1893), "Transcendenz von e und π.", Mathematische Annalen, 43 (2–3): 222–224, doi:10.1007/bf01443647, S2CID 123203471

Gordan, P.

(1873), "Sur la fonction exponentielle.", Comptes rendus de l'Académie des Sciences de Paris, 77: 18–24

Hermite, C.

(1874), Sur la fonction exponentielle., Paris: Gauthier-Villars

Hermite, C.

(1893), "Ueber die Transcendenz der Zahlen e und π.", Mathematische Annalen, 43 (2–3): 216–219, doi:10.1007/bf01443645, S2CID 179177945, archived from the original on 2017-10-06, retrieved 2018-12-24

Hilbert, D.

(1882), "Über die Ludolph'sche Zahl.", Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, 2: 679–682

Lindemann, F.

(1882), "Über die Zahl π.", Mathematische Annalen, 20: 213–225, doi:10.1007/bf01446522, S2CID 120469397, archived from the original on 2017-10-06, retrieved 2018-12-24

Lindemann, F.

Murty, M. Ram; Rath, Purusottam (2014). "Baker's Theorem". . pp. 95–100. doi:10.1007/978-1-4939-0832-5_19. ISBN 978-1-4939-0831-8.

Transcendental Numbers

(1885), "Zu Lindemann's Abhandlung. "Über die Ludolph'sche Zahl".", Sitzungsberichte der Königlich Preussischen Akademie der Wissen-schaften zu Berlin, 5: 1067–1085

Weierstrass, K.

(1990), Transcendental number theory, Cambridge Mathematical Library (2nd ed.), Cambridge University Press, ISBN 978-0-521-39791-9, MR 0422171

Baker, Alan

(1997), "Theta functions and transcendence", The Ramanujan Journal, 1 (4): 339–350, doi:10.1023/A:1009749608672, S2CID 118628723

Bertrand, D.

(2015) [1960], Transcendental and Algebraic Numbers, Dover Books on Mathematics, translated by Boron, Leo F., New York: Dover Publications, ISBN 978-0-486-49526-2, MR 0057921

Gelfond, A.O.

(2009) [1985], Basic Algebra, vol. I (2nd ed.), Dover Publications, ISBN 978-0-486-47189-1

Jacobson, Nathan

"Hermite-Lindemann Theorem". MathWorld.

Weisstein, Eric W.