$K$ is a finite field and $F$ is the set of all functions of $K$ in $K$. In $F$ we define $(f+g)(a)=f(a)+g(a)$ and $(fg)(a)=f(a)g(a)$.
Show that $F$ is isomorphic a $k[x]/I$ for a some ideal $I$ of $k[x]$ , and find this $I$.
Is correct my solution and $I$ is only this $\ker(\Phi)$? What influences $K$ be field and not just ring.
$\Phi$ is homomorphism , because $(f+g)(a)=f(a)+g(a)$ and $(fg)(a)=f(a)g(a)$.
if $k[x]/I$ is isomorphic a $\Phi$ => $\ker(\Phi)= I = \{p(x) \in k[x] \mid \Phi(p(x))=0\}$ and $\Phi$ is homomorphism
So $k[x]/I$ is isomorphic a $\Phi$ , and $I = \ker(\Phi)$