# Prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using a theorem.

Prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using Gelfond-Schneider’s theorem.

I’m interested in this problem because I knew that ${\sqrt2}^{\sqrt2}$ is a transcendental number by Gelfond-Schneider’s theorem. I’ve tried to prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using the Gelfond-Schneider’s theorem, but I’m facing difficulty. I need your help.

I crossposted to MO:
https://mathoverflow.net/questions/138247