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

Question

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

Answer 1

I'm posting an answer just to inform that the question has received an answer by Mark Sapir on MO.

https://mathoverflow.net/questions/138247/prove-that-sqrt2-sqrt2-is-an-irrational-number-without-using-a-theorem