I am trying to show that for any polynomial $f(x)$ in $K[x]$ where $K$ is an algebraic extension of $F$, there is some polynomial in $F[x]$ that has $f$ as a factor.
I'm trying to show this by saying that $(f)$ intersects with $F[x]$ but am not sure how to approach it.
There is a matrix $A$ over $K$ with minimum polynomial $f$ (for instance the companion matrix). As $K$ is a finite extension of $F$ we can regard $A$ as a map from a finite-dimensional vector space over $F$ to itself, so as a square matrix $B$ over $F$ (of $|K:F|$ times the size of $A$). Let $g$ be the characteristic polynomial of $B$. I claim that $f\mid g$. Considering $A$ and $B$ as endomorphisms of finite-dimensional vector spaces, we see that as $g(B)=0$, then $g(A)=0$ and as $f$ is the minimum polynomial of $A$ then $f\mid g$.