 Lindemann–Weierstrass theorem

For distinct algebraic numbers ${\alpha }_1,\dots ,{\alpha }_s$, the numbers $e^{{\alpha }_1},\dots ,e^{{\alpha }_s}$ are linearly independent over $\overline{\mathbb{Q}}$. Equivalently, exponentials of numbers linearly independent over $\mathbb{Q}$ are algebraically independent over $\overline{\mathbb{Q}}$. The special case of a nonzero algebraic exponent gives a transcendental exponential. In particular $e^{i\pi }=-1$ forces $\pi$ to be transcendental.

The Lindemann–Weierstrass theorem is a fundamental result in complex analysis and transcendental number theory.