## Characterize the elements of F[x]/(p(x)) with F a field

Let $F$ be a field, and let $p(x) \in F[x]$ be a polynomial of degree $n \geq 1$. Let bars denote passage to $F[x]/(p(x))$. Prove that for every $q(x) \in F[x]$ there is a unique $q_0(x) \in F[x]$ such that $q_0(x)$ has degree less than $n$ and $\overline{q(x)} = \overline{q_0(x)}$.

Recall that $F[x]$ is a Euclidean domain. By the division algorithm, then, there exist $a(x)$ and $q_0(x)$ such that $q(x) = a(x)p(x) + q_0(x)$ and $q_0$ has degree less than $n$. Certainly then $\overline{q(x)} = \overline{q_0(x)}$. Moreover, we know $q_0$ is unique from the proof that $F[x]$ is a Euclidean domain.