 Solution

Put $E=F(a,b)$. The tower law for field extensions shows that both $m$ and $n$ divide $[E:F]$. Since $m,n$ are coprime, their product divides $[E:F]$, so $[E:F]\ge mn$.

The minimal polynomial of $b$ over $F$ has degree $n$ and is still an annihilating polynomial over $F(a)$. The minimal polynomial over $F(a)$ divides it, soThe upper and lower bounds agree:Coprimality is the step forcing the lower bound to equal the product; without it the two generated fields can overlap substantially.