Schanuel's conjecture

For a subfield $F\subseteq\mathbb C$, write $\operatorname{trdeg}_{\mathbb Q}F$ for its transcendence degree over $\mathbb Q$: the largest number of elements of $F$ that are algebraically independent over $\mathbb Q$.

Conjecture (Schanuel). Let $n\ge1$ and let $z_1,\dots,z_n\in\mathbb C$ be linearly independent over $\mathbb Q$. Then

$$\operatorname{trdeg}_{\mathbb Q}\,\mathbb Q\bigl(z_1,\dots,z_n,e^{z_1},\dots,e^{z_n}\bigr)\ \ge\ n.$$

A solution must prove this for every $n$ and every $\mathbb Q$-linearly independent tuple. The bound is sharp: if $z_1,\dots,z_n$ are $\mathbb Q$-linearly independent algebraic numbers, the Lindemann–Weierstrass theorem gives transcendence degree exactly $n$. The cases $n=1$ and, more generally, tuples of algebraic $z_i$ are theorems. The conjecture is open for $n=2$.

Background

Schanuel stated the conjecture in the 1960s. It first appeared in print in Lang's book on transcendental numbers [1]. It contains most known and expected results on the transcendence of values of the exponential function as special cases. Taking $z_i$ algebraic gives the Lindemann–Weierstrass theorem. Taking $z_i=\log\alpha_i$ for algebraic $\alpha_i$ predicts that $\mathbb Q$-linearly independent logarithms of algebraic numbers are algebraically independent, a strong form of Baker's theorem [2]. Taking $(z_1,z_2)=(1,i\pi)$ gives that $e$ and $\pi$ are algebraically independent. It is not known whether $e+\pi$ is irrational.

The conjecture is also central in model theory. Macintyre and Wilkie proved that the first-order theory of $(\mathbb R,+,\cdot,\exp)$ is decidable if the real version of Schanuel's conjecture holds [5]. Zilber constructed "pseudo-exponential" fields that satisfy Schanuel's conjecture and conjectured that $(\mathbb C,+,\cdot,\exp)$ is one of them [6].