Gelfond-Schneider theorem

In this masters thesis we prove by contradiction the irrationality of the numbers e, π 2 , and √n m, where m, n ∈ N and √n m ∈/ N. Alongside we also prove the transcendence of the numbers e and π, which have distinctly different proofs, as the proof for the transcendence of the number π makes use of the theory of elementary symmetric polynomials. In the last chapter, we also prove the main theorem of this work, which is the Gelfond-Schneider theorem on the transcendence of numbers of the form α β , where α 6= 0, 1 and β is irrational. This work is one of the rare translations of this theorem in Slovene; the only other translation appears in an undergraduate thesis from 1978 [6]. We conclude the thesis by listing several consequences of the theorem and a number of authors who advanced the work in transcendental number theory