# Abstract Algebra problem (related to Rings of Polynomails

1. Dec 9, 2008

### JamesF

We'll I made it through another semester, but it seems that I am completely stuck on the last problem of the last homework assignment. I've made a little progress, but I'm really having trouble understanding the question. Perhaps someone on these forums will have some insight

1. The problem statement, all variables and given/known data
Let F be a field. An element $$\phi \in F^F$$ is a polynomial function on F if there exists an $$f(x) \in F[x] \text{ s.t. } \phi(a) = f(a) \; \forall a \in F$$

a) Show that the set $$P_F$$ of all polynomial functions on F form a subring of $$F^F$$

b) Show that the ring $$P_F$$ is not necessarily isomorphic to F[x] (Hint: show that if F is a finite field, then $$P_F$$ and F[x] don't have the same number of elements)

2. Relevant equations
In this problem, $$F^F$$ refers to the set of all functions mapping F to F, and F[x] refers to the ring of polynomials with coefficients in F.

3. The attempt at a solution

Part b) doesn't seem so difficult. If we let $$F = \mathbb{Z}_2$$, then there are only 4 functions in all since there are only two possible values in the domain and range. But $$\mathbb{Z}_2 [x]$$ has infinitely many elements, so they cannot be isomorphic.

Part a) though has me stumped. We have that $$P_F \subseteq F^F$$, so for any $$\phi \in P_F, \longrightarrow \phi: F \rightarrow F$$. If we let $$\phi, \psi \in P_F \; f(x), g(x) \in F[x]$$, then we have that $$\phi(a) = f(a), \; \psi(a) = \g(a)$$.

Am I understanding this correctly? How can I demonstrate that it satisfies the ring axioms? Just directly apply them and see if it works? Or am I missing something?