Homomorphisms between fields are injective.

Suppose $f(a) = f(b)$, then $f(a-b) = 0 = f(0)$. If $u = (a-b) \ne 0$, then $f(u)f(u^{-1}) = f(1) = 1$, but that means that $0 f(u^{-1}) = 1$, which is impossible. Hence $a - b = 0$ and $a = b$.

A field homomorphism must in particular be a ring homomorphism, so its kernel is an ideal. The only ideals of a field are the zero ideal and the field itself.