Let $F$ be a field. An element $\phi$ of $F^F$ is a polynomial function on $F$, if there exists $f\in F[x]$ such that $\phi (a)=f(a)$ for all $a \in F$.
Show that the set $P_F$ of all polynomial functions on $F$ forms a subring of $F^F$.
I know we are suppose to use homomorphisms but I don’t understand how or why. I’m just really struggling with this proof as a whole, so the more detail the better. Thank you!
Hints: Can you define a map from $F[x]\to F^F$? (That is, for any element of $F[x]$, what function from $F$ to itself does it correspond to?)
Next, once you've done that, you should show that the map is a homomorphism of rings. What are the operations in each of these rings?
Finally, what do you know about the image of a ring under a ring homomorphism?