It is an open problem to prove that $\pi$ and $e$ are algebraically independent over $\mathbb{Q}$.
- What are some of the important results leading toward proving this?
- What are the most promising theories and approaches for this problem?
People in model theory are currently studying the complex numbers with exponentiation. Z'ilber has an axiomatisation of an exponential field (field with exponential function) that looks like the complex numbers with exp. but satisfies Schanuel's conjecture. He proved that there is exactly one such field of the size of $\mathbb C$. I would find it odd if Z'ilber's field turned out to be different from the complex numbers.
By results of Wilkie, the reals with exponentiation are well understood, and the complex numbers with exponentiation is in some way the next step up. The model theoretic frame work (o-minimality) that works for the reals with exp. fails for the complex numbers, but there might be a similar theory that works for the complex field with exponentiation.
Schanuel's conjecture would imply this result. It states that if $z_1, \ldots, z_n$ are linearly independent over $\mathbb{Q}$, then $\mathbb{Q}(z_1, \ldots, z_n, e^{z_1}, \ldots, e^{z_n})$ has transcendence degree at least $n$ over $\mathbb{Q}$. In particular, if we take $z_1 = 1$, $z_2 = \pi i$, then Schanuel's conjecture would imply that $\mathbb{Q}(1, \pi i, e, -1) = \mathbb{Q}(e, \pi i)$ has transcendence degree 2 over $\mathbb{Q}$.
There is a proof of the algebraic independence of $\pi$ and $e^\pi$ in Introduction to Algebraic Independence Theory and a detailed exposition of methods created in last the 25 years although I have not read it.