Studying the $F[x]/\langle p(x)\rangle$ when $p(x)$ is any degree.

Question

Let $F$ be a field, $p(x)\in F[x]$. If $p(x)=0$, then $F[x]/\langle p(x)\rangle=F[x]/\langle 0\rangle=\bigl\{\{a\}\mid a\in F[x]\bigr\}$, a set of sets of one element. If $\deg p(x)=0$, then $F[x]/\langle p(x)\rangle=\{F[x]\}$, a set with one element. Am I correct? And if $\deg p(x)\geq1$, under what circumstances will $F[x]/\langle p(x)\rangle$strictly larger than $F$ (view $F$ as the isomorphic copy in it)?