= Lindemann–Weierstrass theorem
{c}
For distinct <algebraic numbers> $\alpha_1,\ldots,\alpha_s$, the numbers $e^{\alpha_1},\ldots,e^{\alpha_s}$ 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.
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