“Prove” that 1 is transcendental.

“Proof”:

The Gelfond–Schneider theorem states that if $\displaystyle \alpha$ and $\displaystyle \beta$ are algebraic numbers such that $\displaystyle \alpha\ne0,\,1$ and $\displaystyle \beta\notin\mathbb Q,$ then $\displaystyle \alpha^\beta$ is a transcendental number.

Hence, taking $\displaystyle \alpha=-1$ and $\displaystyle \beta=\sqrt2,$ we have that $\displaystyle (-1)^{\sqrt2}$ is transcendental.

But $\displaystyle (-1)^{\sqrt2}=(-1)^{2\frac{\sqrt2}2}=\left((-1)^2\right)^{\frac{\sqrt2}2}=1^{\frac{\sqrt2}2}=1.$

Hence 1 is transcendental.

Explain what is wrong with the above “proof”.