Are $\pi$ and $e$ algebraically independent?

Update Edit : Title of this question formerly was "Is there a polynomial relation between $e$ and $\pi$?"

Is there a polynomial relation (with algebraic numbers as coefficients) between $e$ or $\pi$ ? For example does there exists algebraic numbers $a_1,a_2,..,a_n$ s.t. $$a_n e^n + a_{n-1}e^{n-1}+\cdots+a_0e^0 = \pi$$ or $$a_n \pi^n + a_{n-1}\pi^{n-1}+\cdots+a_0\pi^0 = e$$

Solution 1:

According to Wikipedia, this is an open problem (as of $17$ years ago, anyway). A common phrase to describe the question (which will help with searches) is "are $\pi$ and $e$ algebraically independent".

an important related problem is the validity of Schanuel's conjecture.

A related thread from over at MO