Gelfond–Schneider theorem

Using the theorem of Gelfond -Schneider was the first time a large class of transcendental numbers are generated. He was away a year later, first proved in 1934 by the Russian mathematician Alexander Gelfond and independently of Theodor Schneider. The sentence answered Hilbert's seventh problem.

Statement of the theorem

Let and algebraic numbers ( with ). is not rational beyond.

Then says the set of Gelfond - Schneider:

Complex Numbers

For complex numbers, and also may be used. Then we have. The complex logarithm is uniquely determined only up to multiples of. The sentence is correct for any choice of the branch of the logarithm.

Applications

The transcendence of the following numbers follows directly from the set:

• The Gelfond -Schneider constant and
• The Gelfond - constant since. Note that not a rational number.
• The real number, since