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(π)Q(π) are transcendental, for any non-constant polynomials P(x),Q(x)∈Q¯[x]. The second is that π≠αβ, for any algebraic numbers α and β. 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.

Keywords

• Schanuel’s Conjecture
• Gelfond–Schneider Theorem
• Hermite–Lindemann Theorem
• algebraic independence
• transcendence degree
• power tower