You may have in your course something (first isomorphism theorem?) like: if is a ring morphism between two rings and thenwe have to prove that Im(alpha) is field if and only if Ker(alpha) is a maximal ideal of R. But I don't know how to prove it.

The fact that for any ring the quotient of by one of its ideal is a field iff is a maximal ideal will allow you to conclude.

ii) So a way to prove that is a field is to show that is surjective homomorphism (i.e. ) and that is a maximal ideal of

Therefore the first question is : what is ?