Open Problem Garden

Importance: Outstanding ✭✭✭✭

Author(s):

Subject:	Number Theory
	» Analytic Number Theory

Keywords:

algebraic independence

Recomm. for undergrads: no

Posted	by:	Charles
	on:	July 8th, 2008

Conjecture Given any $n$ complex numbers $z_1,...,z_n$ which are linearly independent over the rational numbers $\mathbb{Q}$ , then the extension field $\mathbb{Q}(z_1,...,z_n,\exp(z_1),...,\exp(z_n))$ has transcendence degree of at least $n$ over $\mathbb{Q}$ .

Schanuel's Conjecture implies the algebraic independence of $\pi$ and $e$ , as well as a positive solution to Tarski's exponential function problem.