Lindemann–Weierstrass theorem

The set of Lindemann - Weierstrass is a number theoretical result about the non-existence of zeroes at certain Exponentialpolynomen, from which, for example, the transcendence of Euler's number and the wave number follows. It is named after the two mathematicians Carl Louis Ferdinand von Lindemann and Karl Weierstrass.

Statement

Be pairwise distinct algebraic numbers and algebraic numbers are arbitrary, although not all are. Then:

This very general theorem proved by Lindemann, to show the much weaker results of the transcendence of the Euler's number and the circle number. In the 1960s, a generalization of this theorem was formulated as a conjecture by Stephen Schanuel, see conjecture of Schanuel.

Shortly after the proof of the theorem of Lindemann - Weierstrass David Hilbert presented a significantly simplified proof for the special cases of the transcendence of the numbers of subjects and from which, in turn, can be inferred also the general rate.

Conclusions

These results follow directly from the above theorem.

Transcendence of e

Would be an algebraic number, so there were not all zero, so that

What would be an obvious contradiction to the above result.

Transcendence of π

In order to show the transcendence of the circle number, we first assume that an algebraic number. Since the set of algebraic numbers forms a body, it would also be algebraically (referred to here is the imaginary unit ).

If we choose now and, we obtain the set of Lindemann - Weierstrass and Euler's identity the contradiction

This shows that our assumption was false, so the loop number must be transcendent.