G. Bergman gave a series of exercises leading to the proof of Luroth's Theorem: https://math.berkeley.edu/~gbergman/grad.hndts/Luroth.ps
My concern lays in problem (7) and (8).
$L$ is any intermediate field: $k \subsetneq L \subset k(t)$, where $t$ is transcendental over $k$. $u=P(t)/Q(t)$ is an element of $L-k$.
Please refer to the pdf file for other details.
Show that if $P(x)-uQ(x)$ is divisible in $L[x]$ by an element of $k[x]$, then this element must divide both $P(x)$ and $Q(x)$.
how to prove it bothers me; my thought is like this: $ \{1,u\} \subset B$, where $B$ is a basis of $L$ over $k$. Then $L[x] = k[u,..][x]$. After we embed $k[x]$ into $L[x]$, $u$ clearly is not divisible by any element from $k[x]$ then $P(x)$ and $Q(x)$ has to be divided by the element respectively. Am I correct?
In addition, is there any theorem leading to such result that if $t$ has an identical minimal polynomial over $k(u)$ and $L$ while $k(u) \subset L$, then $L=k(u)$?