Schanuel's Conjecture

Importance: Outstanding ✭✭✭✭
Author(s): Schanuel, Stephen
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.


* 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.

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