Schanuel's conjecture (statement)

Conjecture. If $\alpha_1,\ldots,\alpha_n$ are $\mathbb{Q}$-linearly independent complex numbers, then the field $\mathbb{Q}(\alpha_1,\ldots,\alpha_n,e^{\alpha_1},\ldots,e^{\alpha_n})$ has transcendence degree at least $n$ over $\mathbb{Q}$.

Consequences.
- Lindemann–Weierstrass theorem (special case).
- Algebraic independence of $e$ and $\pi$ (take $n=2$, $\alpha_1=1$, $\alpha_2=i\pi$).
- Gelfond–Schneider theorem.

Conclusion. Schanuel's conjecture is a vast generalization; it remains open and is one of the deepest conjectures in transcendence theory.