Divisors of irreducible polynomials over a field

Let F be a field and suppose p(x) \in F[x] is irreducible. If q(x)|p(x), what can we say about q(x)?

Note that the units in F[x] are precisely the nonzero constant polynomials. If q(x)|p(x), then we have p(x) = q(x)t(x) for some t(x) \in F[x]. Since p(x) is irreducible, either q(x) \in F^\times is a unit, or q is a constant multiple of p(x).

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