The following problem is very well-known among algebraic geometers:
Does there exist a cubic 4-fold that is not rational?
It's probably not well-known outside of algebraic geometry, even though it can easily be explained in every elementary terms:
Does there exist a polynomial equation $F$ of degree three in five variables with the following property: Let $X \subset \mathbb C^5$ be the solution set of $F = 0$. Then there exists no chart $U \subset \mathbb C^4, \phi \colon U \to X$ such that $\phi$ is defined by rational functions (i.e., quotients of polynomials).
