Extension fields of coprime degrees

Question

Let $u,v$ be algebraic over the field $K$, such that $[K(u):K]=n$ and $[K(v):K]=m$. Show that if $\gcd(m,n)=1$, then $[K(u,v):K]=mn$.

Once $[K(u,v):K]=[K(u,v):K(u)][K(u):K]=n[K(u,v):K(u)]$, I'm trying to show that $[K(u,v):K(u)]=m$.

Answer 1

Note $r=[K(u,v):K(u)]$ and $s=[K(u,v):K(v)]$. As $K \subset K(v)$ we have $s \le n$.

We also have $rn=sm$ and if $\gcd(m,n)=1$, $n$ divides $s$ (by Euclid lemma). So finally $n=s$ and also $m=r$.