Let , ..., be linearly independent over the rationals , then
has transcendence degree at least over . Schanuel's conjecture implies the Lindemann-Weierstrass theorem and Gelfond's theorem. If the conjecture is true, then it follows that and are algebraically independent. Macintyre (1991) proved that the truth of Schanuel's conjecture also guarantees that there are no unexpected exponential-algebraic relations on the integers (Marker 1996).
At present, a proof of Schanuel's conjecture seems out of reach (Chow 1999).