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.

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