You need to show that satisfy all the properties of a ring. For example, is commutative because . Also is commutative since . You also need to show distributive laws . And you need to show that there is an identity element for and inverses for it as well.

Is there an element so that ? Yes! Take then since it will follow . Thus, is the identity element for . However, does every element have an inverse i.e. for any there is so that . And the answer is of course not.Is P(x) with this ringstructure a field?

Not sure I understand what exactly you trying to show here.3.) Proof that P(X) is a Vector Space over the Galois-field.