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.