Schanuel's Conjecture

Let $\lambda_1$, ..., $\lambda_n\in\Bbb{C}$ be linearly independent over the Rationals $\Bbb{Q}$, then

$$\Bbb{Q}(\lambda_1, \ldots, \lambda_n, e^{\lambda_1}, \ldots, e^{\lambda_n})$$

has Transcendence degree at least $n$ over $\Bbb{Q}$. Schanuel's conjecture is a generalization of the Lindemann-Weierstraß Theorem. If the conjecture is true, then it follows that $e$ and $\pi$ are algebraically independent. Mcintyre (1991) proved that the truth of Schanuel's conjecture also guarantees that there are no unexpected exponential-algebraic relations on the Integers $\Bbb{Z}$ (Marker 1996).

See also Constant Problem

References

Macintyre, A. ``Schanuel's Conjecture and Free Exponential Rings.'' Ann. Pure Appl. Logic 51, 241-246, 1991.

Marker, D. ``Model Theory and Exponentiation.'' Not. Amer. Math. Soc. 43, 753-759, 1996.