TITLE: Isomorphism between fields
QUESTION [6 upvotes]: Suppose that $F$ and $G$ are two fields and $F[x]$ and $G[x]$ the polynomial rings in $x$. We know that the equivalence classes of the fractions of elements of $F[x]$ is a field $F(x)$ and the same is for $G(x)$.

If $F[x]$ and $G[x]$ are isomorphic as rings and/or $F(x)$ and $G(x)$ are isomorphic as fields, what can we say about $F$ and $G$?

I remember that I've read that we cannot prove that $F$ and $G$ are isomorphic, but I don't remember why and I suppose that the proof is not simple.

REPLY [5 votes]: Here's one part: Assume $\phi\colon F[x]\to G[x]$ is an isomorphism of rings with inverse $\psi$. For $a\in F^\times$, we must have $\phi(a)\in G[x]^\times =G^\times$. As also $\phi(0)=0$, we have that $\phi|_F$ is a ring homomorphism $F\to G$ with inverse $\psi|_G\colon G\to F$, in other words, $F$ and $G$ are isomorphic.