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| Lindemann–Weierstrass theorem | |
|---|---|
| Name | Lindemann–Weierstrass theorem |
| Field | Number theory |
| Proved | 1882 |
| By | Ferdinand von Lindemann, Karl Weierstrass |
Lindemann–Weierstrass theorem The Lindemann–Weierstrass theorem is a central result in Number theory asserting transcendence of certain exponential values and linear independence of algebraic numbers under exponentiation. It implies classical conclusions about constants studied by Leonhard Euler, Carl Friedrich Gauss, Évariste Galois, Joseph Liouville, Srinivasa Ramanujan, and later investigators such as David Hilbert, André Weil, Alexander Grothendieck, and Kurt Mahler. The theorem connects themes from work by Ferdinand von Lindemann, Karl Weierstrass, Charles Hermite, Georg Cantor, Richard Dedekind, and developments in Algebraic number theory, Complex analysis, and Transcendental number theory.
The theorem states that if α1, α2, ..., αn are distinct algebraic numbers (from fields studied by Évariste Galois and Richard Dedekind) then e^{α1}, e^{α2}, ..., e^{αn} are linearly independent over the algebraic numbers; equivalently, for any algebraic numbers β1, β2, ..., βn not all zero, the linear combination β1 e^{α1} + β2 e^{α2} + ... + βn e^{αn} ≠ 0. This formulation builds on earlier results by Charles Hermite on e and on methods connected to Karl Weierstrass’s analytic theory and Ferdinand von Lindemann’s work proving the transcendence of π after Georg Cantor’s set-theoretic foundations and contemporaneous advances by Leopold Kronecker. The statement is often presented in the language of field extensions and Linear algebra over number fields considered by David Hilbert and Emmy Noether.
The origin traces to Charles Hermite’s 1873 proof of the transcendence of e, which influenced Ferdinand von Lindemann to prove π’s transcendence in 1882 by showing e^{α} is transcendental for nonzero algebraic α; Lindemann cited methods from Karl Weierstrass and ideas resonant with work of Adolf Hurwitz, Carl Gustav Jacobi, and Sophus Lie regarding analytic functions. Later refinements and a more general proof attributing the stronger linear independence formulation are associated with Karl Weierstrass and were systematized in the context of Transcendental number theory by Theodor Schneider and Thue–Siegel–Roth theorem proponents such as Kurt Mahler and Alan Baker. Modern expositions employ techniques from Complex analysis like the theory of entire functions used by Henri Poincaré, constructions of auxiliary functions informed by Joseph Liouville and Srinivasa Ramanujan, and algebraic concepts from Évariste Galois and Richard Dedekind. Later proofs and simplifications appear in works by Alan Baker, Kurt Mahler, Theodor Schneider, Wolfgang Steinitz, and researchers at institutions like École Normale Supérieure, University of Göttingen, and Princeton University.
A primary consequence is the transcendence of π, settling questions raised in problems linked to Greek mathematics and the classical quadrature of the circle pursued by figures from Archimedes to Isaac Newton and Carl Friedrich Gauss. The theorem implies that for any nonzero algebraic α, e^{α} is transcendental, affecting constants investigated by Leonhard Euler, Johann Lambert, and Adrien-Marie Legendre. Applications appear in results on algebraic independence used by André Weil and Alexander Grothendieck in motives and periods, influence on Diophantine approximation studied by Paul Erdős, Kurt Mahler, and Alan Baker, and consequences for special values of exponential and elliptic functions relevant to Niels Henrik Abel, Carl Gustav Jacob Jacobi, and Srinivasa Ramanujan. The theorem also feeds into transcendence criteria utilized by researchers at Institute for Advanced Study and by contemporary mathematicians such as Michel Waldschmidt and David Masser.
Classical special cases include Hermite’s result that e is transcendental (proved by Charles Hermite), Lindemann’s proof that π is transcendental (by Ferdinand von Lindemann), and corollaries like the transcendence of e^{r} for nonzero rational r (connected to work by Joseph Liouville and Évariste Galois). Specific evaluations impacting constants from Leonhard Euler include the nonalgebraicity of expressions like e^{π}, e^{π i} in contexts explored by Augustin-Louis Cauchy and Bernhard Riemann in complex function theory. Instances where the theorem applies include linear independence statements for exponentials of algebraic numbers studied by David Hilbert in his problems and by Alan Baker in his investigations into logarithms of algebraic numbers.
Generalizations extend to results by Theodor Schneider and Gelfond–Schneider theorem proving transcendence of values like a^{b} for algebraic a ≠ 0,1 and irrational algebraic b; further extensions by Alan Baker give linear forms in logarithms, and work by Kurt Mahler explores p-adic analogues connected to Hermann Minkowski and Ernst Steinitz. Conjectural frameworks include the Schanuel's conjecture formulated by Stephen Schanuel, which would generalize Lindemann–Weierstrass to broader algebraic independence statements and link to research by Pierre Deligne, David Mumford, and Grothendieck on periods and motives. Related fields of study involve transcendental number theory advanced at institutions such as Université Paris-Sud, ETH Zurich, and Cambridge University by researchers like Michel Waldschmidt, David Masser, Alan Baker, and Yuri Nesterenko.
Category:Theorems in number theory