Theorem by Lindemann-Weierstrass

The Lindemann-Weierstrass theorem is a number theoretical result on the non-existence of nulls at certain Exponentialpolynomen, from which for example, the transcendental the Euler's number ${\displaystyle e}$ , and the circuit number ${\displaystyle \pi }$ follows. It is named after the two mathematicians Carl Louis Ferdinand von Lindemann and Karl Weierstrass .

statement

Given a (finite) set of algebraic numbers , the images of these numbers under the exponential function are linearly independent over the field of the algebraic numbers.

This very general theorem was (partially) proven by Lindemann in 1882, based on the Hermitian matrix , in order to show the transcendence of Euler's number ${\displaystyle e}$ and the number of circles ${\displaystyle \pi }$ . Although he suggested extensions, these remained unpublished, so that Weierstrasse completed it in 1885. Both works together form the proof, so that the sentence was named "Theorem from Lindemann-Weierstrass".

In 1893, however , David Hilbert presented a clearly simplified proof by contradiction for the special cases of the transcendence of numbers and , from which the general theorem can in turn be deduced. ${\displaystyle e}$ ${\displaystyle \pi }$

In the 1960s Stephen Schanuel formulated a generalization of this theorem as a conjecture, see Schanuel's conjecture .

Inferences

These results follow directly from the sentence above.

Transcendence of e

If it were an algebraic number , it would be the zero of a normalized polynomial with rational coefficients. So there would be rational numbers such that ${\displaystyle e}$ ${\displaystyle e}$ ${\displaystyle \beta _{0},\dots ,\beta _{n-1}}$

${\displaystyle e^{n}+\beta _{n-1}e^{n-1}\cdots +\beta _{1}e^{1}+\beta _{0}e^{0}=0}$.

Thus the first powers of e would be linearly dependent over (and thus also over ) in contradiction to Lindemann-Weierstrass's theorem. ${\displaystyle n+1}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle {\overline {\mathbb {Q} }}}$

Transcendence of π

To show the transcendence of the circle number , let's first assume that is an algebraic number. Since the set of algebraic numbers forms a field , it should also be algebraic ( here denotes the imaginary unit ). But now is ${\displaystyle \pi }$ ${\displaystyle \pi }$ ${\displaystyle \pi i}$ ${\displaystyle i}$

${\displaystyle e^{\pi i}+e^{0}=-1+1=0}$

contrary to linear independence of and . ${\displaystyle e^{\pi i}}$ ${\displaystyle e^{0}}$

This shows that our assumption was wrong, so the circle number must be transcendent. ${\displaystyle \pi }$

literature

• Charles Hermite: Sur la fonction exponential. In: Comptes Rendus Acad. Sci. Paris 77 , (1873), pp. 18-24.
• Charles Hermite: Sur la fonction exponential. Gauthier-Villars, Paris (1874).
• Ferdinand Lindemann: About the Ludolph number. In: Session reports of the Royal Prussian Academy of Sciences in Berlin 2 (1882), pp. 679–682.
• Ferdinand Lindemann: About the number . ${\displaystyle \pi }$In: Mathematische Annalen 20 (1882), pp. 213-225.
• Karl Weierstrass: On Lindemann's treatise. "About the Ludolph number". In: Meeting reports of the Royal Prussian Academy of Sciences in Berlin 5 (1885), pp. 1067-1085.
• David Hilbert: About the transcendence of the numbers e and . ${\displaystyle \pi }$In: Mathematische Annalen 43 (1893), pp. 216-219.

Individual evidence

1. ↑ David Hilbert: About the transcendence of numbers and ${\displaystyle e}$ ${\displaystyle \pi }$ , digitized , also Wikibooks