Consider an integrally closed domain $A$, and a field automorphism $\phi$ of its fraction field $K$. Is it true that $\phi$ sends $A$ in $A$?
I tried using Hartogs's theorem $$A=\bigcap_{p\in \rm Spec A} A_p$$ but it didn't help me.
Probably it is something obvious, and one does not even need Hartogs. Thank you in advance!