field homomorphisms fix prime subfields
Theorem.
Let and be fields having the same prime subfield and be a field homomorphism. Then fixes .
Proof.
Without loss of generality, it will be assumed that is either or .
Since is a field homomorphism, , , and, for every , .
Let and be the characteristic of . Then
This the proof in the case that is prime.
Now consider . Let . Then there exist with such that . Thus, . Therefore, . Hence, fixes .∎