TITLE: is an automorphism on the prime field the identity map?
QUESTION [6 upvotes]: Let $F$ be a field and f an automorphism on K. Is $f$ the the identity map on the prime field of $F$? I feel it should follow from the fact that the prime field is either $\mathbb{Q}$ or $F_p$, but I cannot find a proof.
Can someone help me out?

REPLY [7 votes]: More generally let $f,g : K \to L$ be two field homomorphisms and let $P$ be the prime field of $K$, i.e. the unique smallest subfield of $K$. Since $\{a \in K : f(a)=g(a)\}$ is a subfield of $K$ (check it!), it has to contain $P$. This means $f|_P = g|_P$.