Schanuel's Conjecture

Given complex numbers z₁,…,\(z_{n}\) linearly independent over \mathbb{Q}, the field \mathbb{Q}(z₁,…,\(z_{n}\), \(e^{z₁}\),…,\(e^{z_{n}}\)) 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.