TITLE: Is $\mathbb Q$ a quotient of $\mathbb R[X]$?
QUESTION [6 upvotes]: Is there some ideal $I \subseteq \mathbb R[X]$ such that $\mathbb R[X]/I \cong \mathbb Q$?
$I$ is clearly not a principal ideal.

REPLY [12 votes]: $\Bbb R[X]/I$ is necessarily a vector space over $\Bbb R$, so it is either $\{0\}$ or it is uncountable. It cannot be isomorphic to the nonzero countable $\Bbb Q$.