Let $K$ be a finite extension of a field $F$, and let $f(x)$ be in $K[x]$. Prove that there is a nonzero polynomial $g(x)$ in $K[x]$ such that $f(x)g(x)$ is in $F[x]$.
Should I do this by induction on the degree of $f(x)$?
Obviously if $n=0$, then $g(x)=1/f(x)$
Let $f(x) = a_nx^n+...a_1x+a_0$ then I know that there exists a h(x) so that $(f(x)-a_nx^n)h(x)$ is in $F[x]$. I want now to find a $g(x)=h(x)+i(x)$ so that $f(x)g(x)$ is in $F[x]$. Thus I need to find an $i(x)$ so that $a_nx^nh(x)+i(x)f(x)$ is in $F[x]$. I feel like this is wrong because I have no control over the degrees of $h(x)$.
Any suggestions?