Let $R$ be an integral domain, and let $F$ be the field of fractions of $R$. Then my question is, does every field automorphism of $F$ restrict to a ring automorphism of $R$? By $R$ I mean the natural embedding of $R$ in $F$.
I’m pretty sure that it’s true, but I’m not sure how to prove it.
EDIT: Berci's comment beat me to it!
No, this needn't be true in general. Consider an "ambiguous" $F$ - a field such that there are two different $R_0,R_1\subseteq F$ each of which has $F$ itself as its field of fractions. If $R_0$ and $R_1$ can be "swapped" by an automorphism of $F$, this gives a counterexample.
The simplest example I can think of is $\mathbb{R}[t]$ and $\mathbb{R}[t^{-1}]$ as subdomains of $\mathbb{R}(t)$: there is an automorphism of $\mathbb{R}(t)$ which swaps $t$ and $t^{-1}$, and this clearly doesn't restrict to an automorphism of $\mathbb{R}[t]$ (or $\mathbb{R}[t^{-1}]$).