Schanuel's Conjecture

Given complex numbers \(z_1, \dots, z_n\) linearly independent over \(\mathbb{Q}\), the field \(\mathbb{Q}(z_1, \dots, z_n, e^{z_1}, \dots, e^{z_n})\) has transcendence degree at least \(n\) over \(\mathbb{Q}\).

Posed around 1965 as the natural strengthening of Lindemann–Weierstrass (1880s) and the six-exponentials theorem (Schanuel itself, 1960s). Ax's 1971 theorem proves its differential-function-field analogue — the conjecture is 'Ax–Schanuel for exp', now a foundational local statement in o-minimal model theory (Wilkie, Zilber pseudo-exponentiation). Every route to a proof would also crack long-standing cases of the algebraic independence of e and π.