Homomorphism : kernel of a field only has 0

Question

i'm new here and i'm struggling to prove that for R,S fields and f : R → S

kernel(f) = {0}

Definition of kernel is : kern(f)= { r ∈ R | f(r) = 0 }

I started the proof but stuck at the start :

∀a ∈ R-{0} => f(a) ≠ 0

Thank you for your help !

Answer 1

For $x \neq 0$, $$1_S = f(1_R)=f(xx^{-1}) = f(x) f(x^{-1})$$

so $f(x) \neq 0$.