# Schanuel's Conjecture

 Importance: Outstanding ✭✭✭✭
 Author(s): Schanuel, Stephen
 Subject: Number Theory » Analytic Number Theory
 Keywords: algebraic independence
 Posted by: Charles on: July 8th, 2008

\begin{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 $\mathbb{Q}$. \end{conjecture}

% You may use many features of TeX, such as % arbitrary math (between $...$ and $$...$$) % \begin{theorem}...\end{theorem} environment, also works for question, problem, conjecture, ... % % Our special features: % Links to wikipedia: \Def {mathematics} or \Def[coloring]{Graph_coloring} % General web links: \href [The On-Line Encyclopedia of Integer Sequences]{http://www.research.att.com/~njas/sequences/}

\Def{Schanuel's Conjecture} implies the algebraic independence of $\pi$ and $e$, as well as a positive solution to \OPrefnum[Tarski's exponential function problem]{1790}.

## Bibliography

% Example: %*[B] Claude Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 114. % %[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: \arxiv[The strong perfect graph theorem]{math.CO/0212070}, % Ann. of Math. (2) 164 (2006), no. 1, 51--229. \MRhref{MR2233847} % % (Put an empty line between individual entries)

* indicates original appearance(s) of problem.

### I must agree with the

I must agree with the previous comment. Schanuel's conjecture is likely the most important open problem in Transcendental Number Theory. I realize that this might not be as major a field as the study of "mimic" numbers, but.....

### Re: I must agree with the

Encouraged by the previous comments, I changed the rating of this problem and the "mimic" one. Thanks for the feedback.

### from Gasses

I am just curious why 'importance' is given as 2 stars when (according to wikipedia) "The conjecture, if proven, would subsume most known results in transcendental number theory." Some of these results include results on this page that have greater importance than 2 stars.