May I refer you to:
Can you please explain in detail, problem 1, the inequality:
$[K(u,v): K(u)] \leq n$ ?
Let's see. First the fact that $[K(v): K]=n$, tell us that we can find a minimum polynomial $p(x) \in K[x]$ such that $p(v)=0$, i.e $v$ is algebraic over the field $K$. Now clearly $K(u)$ is a field extension of $K$ therefore we have:
$p(x) \in (K(u))[x]$
Now consider the field extension $K(u,v)$ of $K(u)$. Let $h(x)$ be the minimum polynomial over $(K(u))[x]$ such that $h(u)=h(v)=0$. In particular $h(v)=0$.
We can't claim that $h=p$ even though both are monic irreducible right? because we don't even know if $p$ satisfies $p(u)=0$, all we know is that $p(v)=0$. So all we can say is that $h$ divides $p$ right? (this is the part I'm kinda confused with).
And then $[K(u,v): K(u)] \leq n$. Sorry if this seems trivial to you, just started learning on my own elementary field theory.