Lindemann theorem

The exponential function $e^z$ takes transcendental values (cf. Transcendental number) at algebraic points $z\neq0$ (cf. Algebraic number); proved by F. Lindemann in 1882. The following more general assertion, which was stated without proof by Lindemann and was proved by K. Weierstrass in 1885, is known as the Lindemann–Weierstrass theorem.

Let $a_1,\dots,a_m$ be non-zero algebraic numbers and let $\alpha_1,\dots,\alpha_m$ be pairwise distinct algebraic numbers; then

$$a_1e^{\alpha_1}+\dots+a_me^{\alpha_m}\neq0.$$

This assertion is equivalent to the following: If $\beta_1,\dots,\beta_n$ are algebraic numbers, linearly independent over the field of rational numbers, then the numbers $e^{\beta_1},\dots,e^{\beta_n}$ are algebraically independent.

The method of proving Lindemann's theorem is known as the Hermite–Lindemann method. It is a development of Hermite's method by which he proved in 1873 that $e$ is transcendental, and it is based on the application of the Hermite identity to certain specially constructed polynomials.

From Lindemann's theorem one can deduce that $\pi$ is transcendental, that the problem of squaring the circle has no solution, and also that the functions $\sin z$, $\cos z$ and $\tan z$ take transcendental values for algebraic $z\neq0$, and that $\ln z$ takes transcendental values for algebraic $z\neq0,1$.

References

[1]	A.O. Gel'fond, "Transcendental and algebraic numbers" , Dover, reprint (1960) (Translated from Russian)
[2]	N.I. Fel'dman, A.B. Shidlovskii, "The development and present state of the theory of transcendental numbers" Russian Math. Surveys , 22 : 3 (1967) pp. 1–79 Uspekhi Mat. Nauk , 22 : 3 (1967) pp. 3–81

Comments

D. Hilbert gave a simplified proof of the theorem, which was later polished by a large number of other authors, see [a1]. In 1988, F. Beukers, J.P. Bézivin and Ph. Robba gave a new elementary proof, see [a2].

References

[a1]	A. Baker, "Transcendental number theory" , Cambridge Univ. Press (1975)
[a2]	F. Beukers, J.P. Bézivin, Ph. Robba, "An alternative proof of the Lindemann–Weierstrass theorem" Amer. Math. Monthly (Forthcoming (1989))