## 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 $p(x)$.