Let , ...,
be linearly independent over the Rationals ,
then

has Transcendence degree at least over . Schanuel's conjecture is a generalization of the Lindemann-Weierstraß Theorem. If the conjecture is true, then it follows that and 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 (Marker 1996).

**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.

© 1996-9

1999-05-26