I once heard that in the freshman year of university of the mathematician Kodaira Kunihiko, he solved a problem stating that "the base e of the natural exponential function is not an irrational of the second degree". I'm not sure what "an irrational of the second degree" means. Could the problem otherwise state that "e is not the square root of a non-perfect square rational number"?
Also, I couldn't find any record of his proof, which is considered "perfect". So I've proven it in my own way below. Please let me know if I made any mistake about the below proof and how to understand the problem.
Problem. The base e of the natural exponential function is not the square root of a non-perfect square rational number.
Proof. Let $\mathbb{Q}^*$ be the set of all non-perfect square rational numbers. It's easy to see that $\mathbb{Q^*\subset Q}\backslash\{0,1\}$. Suppose that there exists number $q\in\mathbb{Q^*}$ such that $e=\sqrt q$. Then, $e^2=q\iff\ln e^2=\ln q=2\not\in\mathbb{R\backslash Q}$. Therefore, $q\not\in\mathbb{Q\backslash}\{0,1\}\supset\mathbb{Q^*}$. This gives the contradiction.