Schanuel’s Conjecture and the Transcendence of Power Towers

We give three consequences of Schanuel’s Conjecture. The first is that P(e)Q(e) and P(π)Q(π) are transcendental, for any non-constant polynomials P(x),Q(x)∈Q¯[x]. second π≠αβ, algebraic numbers α β. third the case Gelfond’s conjecture (about transcendence a finite power tower) in which all elements equal.