TITLE: Is $\mathbb{R}$ isomorphic to $\mathbb{R}(x)$?
QUESTION [5 upvotes]: Is $\mathbb{R}$ isomorphic (as a field) to $\mathbb{R}(x)$? (where $x$ is an indeterminate on $\mathbb{R}$)

REPLY [4 votes]: $\mathbb R$ satisfies the condition $\forall t\exists s(t=s^2\lor -t=s^2)$, but $\mathbb R[x]$ does not (consider $t=x$). So they're not isomorphic (in fact they're not even "elementarily equivalent").
{"set_name": "stack_exchange", "score": 5, "question_id": 2527871}