Referring to Exercise 29, let F be a field. An element ∅ of F^F is a polynomial function on F, if there exists f(x) ∈ F[x] such that ∅(a)= f(a) for all a ∈ F. 

a. Show that the set P_F of all polynomial functions on F forms a subring of F^F. 

b. Show that the ring P_F is not necessarily isomorphic to F[x].

Data from Exercise 29

Let R be a ring, and let R^R be the set of all functions mapping R into R. For∅, ψ ∈ R^R, define the sum∅ + ψ by (∅ + ψ)(r) = ∅(r) + ψ(r) and the product ∅ . ψ by (∅ · ψ)(r) = ∅(r)ψ(r) for r ∈ R. Note that · is not function composition. Show that (R^R,+,•) is a ring.