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.