Open problem, Arithmetic and number theory, Algebra, posed 1966

Schanuel's conjecture

Open

If $z_1, z_2, \dots, z_n \in \mathbb{C}$ are linearly independent over $\mathbb{Q}$ , then the field extension $\mathbb{Q}(z_1, \dots, z_n, e^{z_1}, \dots, e^{z_n})$ has transcendence degree at least $n$ over $\mathbb{Q}$ .

Research frontier as of 2026

As of 2026, Schanuel's conjecture remains open for every $n \ge 2$ . While the functional analogue (Ax's theorem, 1971) is completely proved and has been extended to shimura varieties and o-minimal geometry (where Ax–Schanuel theorems drive breakthroughs on the André–Oort and Zilber–Pink conjectures), the number-theoretic conjecture over $\mathbb{Q}$ is far out of reach. When $e^{z_1}, \dots, e^{z_n}$ are algebraic, Baker's theorem (1966) proves linear independence of $z_1, \dots, z_n$ over $\overline{\mathbb{Q}}$ , which yields transcendence degree $\ge 1$ , whereas Schanuel's conjecture demands algebraic independence of $z_1, \dots, z_n$ (transcendence degree $n$ ); even the four-exponentials conjecture (a multiplicative $n = 2$ consequence) remains open.

Best known results

Lindemann–Weierstrass theorem (1882/1885): Schanuel's conjecture holds whenever $z_1, \dots, z_n \in \overline{\mathbb{Q}}$ are algebraic numbers.
Six-exponentials theorem (Lang, Ramachandra 1960s): if $x_1, x_2$ and $y_1, y_2, y_3$ are each $\mathbb{Q}$ -linearly independent, at least one of the six numbers $e^{x_i y_j}$ is transcendental.
Ax's theorem (1971): the differential-field analogue of Schanuel's conjecture holds for formal power series $y_1, \dots, y_n \in \mathbb{C}[[t_1, \dots, t_m]]$ .

Tools and where they stop

Tool	Achieved	Where it stops
Auxiliary polynomials, interpolation determinants, and multiplicity estimates (Baker–Waldschmidt–Philippon)	Proves Baker's theorem, the six-exponentials theorem, and lower bounds on transcendence degree of fields generated by exponentials $e^{x_i y_j}$ (growing like $\frac{mn}{m+n}$ )	The Siegel's-lemma balance between degree and height yields transcendence degree bounds proportional to $\frac{mn}{m+n}$ rather than $n$ , falling short of even the four-exponentials conjecture ( $m = n = 2$ )
Differential algebra and o-minimal geometry (Ax–Schanuel methods)	Proves Schanuel's inequality in full for functions and power series because differentiation $\partial (e^y) = e^y \partial y$ provides a derivation over the base field $\mathbb{C}$	The field of rational numbers $\mathbb{Q}$ has only the trivial derivation $\partial = 0$ , so differential-algebraic proofs cannot be specialized to complex numbers $z_i \in \mathbb{C}$

Open questions

Does Schanuel's conjecture hold for $n = 2$ , and in particular are $\ln 2$ and $\ln 3$ algebraically independent over $\mathbb{Q}$ ?
Does the four-exponentials conjecture hold: if $x_1, x_2$ and $y_1, y_2$ are each $\mathbb{Q}$ -linearly independent, is at least one of the four numbers $e^{x_i y_j}$ transcendental?

References

Serge Lang (1966). Introduction to Transcendental Numbers
James Ax (1971). On Schanuel's conjectures · DOI:10.2307/1970774
Michel Waldschmidt (2000). Diophantine Approximation on Linear Algebraic Groups · DOI:10.1007/978-3-662-11569-5