Let $E$ be an extension of $F$, and let $a, b \in E$ be algebraic over $F$. Suppose that the extensions $F(a)$ and $F(b)$ of $F$ are of degrees $m$ and $n$, respectively, where $(m,n)=1$. Show that $[F(a,b):F]=mn$.
Since $[F(a,b):F]=[F(a,b):F(a)][F(a):F]$ and $[F(a):F]=n$ we have $n|[F(a,b):F]$ with the same argument we prove that $m|[F(a,b):F]$, then $mn|[F(a,b):F]$ and $mn \le [F(a,b):F]$.
My problem is with the converse, I need help.
Thank you