Referring to Exercise 29, let F be a field. An element ϕ of FF is a polynomial function on F, if there exists f(x) ∈ F[x] such that ϕ (a) = f(a) for all a ∈ F.
a. Show that the set PF of all polynomial functions on F forms a subring of FF.
b. Show that the ring PF is not necessarily isomorphic to F[x]. [Hint: Show that if F is a finite field, PF and F[x] don't even have the same number of elements.]
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