Prove or disprove that $2^{\frac{\pi}{2}}$ is an irrational number.

Question

Problem

Prove or disprove that $2^{\frac{\pi}{2}}$ is an irrational number.

My Try

According to our mathematical intuition, we may want to apply Gelfond–Schneider theorem, which states that

if $\alpha,\beta$ are two algebraic numbers, where $\alpha$ dose not equal $0$ or $1$ and $\beta$ is not a rational number, then $\alpha^\beta$ is a transcendental number.

But in fact, it can't work here, because $\beta=\dfrac{\pi}{2}$ here is not an algebraic number at all, which doesn't satisfy the premise of the theorem.

Besides, I have tested this on WolframAlpha. It outputs the consequence as follows

Now, how to go on?

Answer 1

As noted in the comment section, the irrationality of $2^\pi$ is unknown, so is that of $2^{\pi/2}$.