## A fact about polynomials over a perfect field

Let $K$ be an extension of $F$, with $F$ a perfect field. Suppose $p(x) \in F[x]$ has no repeated irreducible factors; prove that $p(x)$ has no repeated irreducible factors in $K[x]$.

Suppose $p$ has no repeated irreducible factors. Since $F$ is perfect, the irreducible factors of $p$ are separable (Prop. 37 on 549 in D&F), and so $p$ is separable. (If not, then it would have a repeated irreducible factor.) That is, $p$ has no repeated roots.

Now if $p$ has a repeated irreducible factor over $K$, then it has repeated roots- a contradiction.