Let F be a finite field of characteristic $p$. Show $f(a) = a^p$ is a ring homomorphism, injective, and surjective Let F be a finite field of characteristic $p$. Show that the function $f:F \to F$ defined by $f(a) = a^p$ is a) a ring homomorphism, b) injective and, c) surjective. 
I tried to approach this problem by proving $f(a+b) = f(a) + f(b), f(ab) =f(a)f(b) $ for ring homomorphism. 
But I don't know how to solve b)and c). Could you please help me to solve it ? Thank you!!!

Hint: a field homomorphism is always injective, and an injective map from a finite set to itself is always surjective.
