Gelfond’s theorem

Let $\alpha$ and $\beta$ be algebraic over $\mathbb{Q}$, with $\beta$ irrational and $\alpha$ not equal to 0 or 1. Then ${\alpha }^{\beta }$ is transcendental over $\mathbb{Q}$.

This is perhaps the most useful result in determining whether a number is algebraic or transcendental.

The theorem is also known as the Gelfond-Schneider Theorem.