Schanuel's Conjecture and the Transcendence of Power Towers

Abstract

We give three consequences of Schanuel's Conjecture. The first is that P(e)(Q(e)) and P(pi)(Q(pi)) are transcendental, for any non-constant polynomials P(x),Q(x) is an element of Q vertical bar x vertical bar. The second is that pi not equal alpha(beta), for any algebraic numbers alpha and beta. The third is the case of the Gelfond's conjecture (about the transcendence of a finite algebraic power tower) in which all elements are equal.