$L/k$ and $K/k$ field extensions of $k$ with $[L:k]=m,[K:k]=n, (m,n)=1\implies L\cap K=k$.
The first thing I taught is that if $\alpha\in L,\beta\in K$ are algebric over $k$, the degrees of the minimal polynomials must not divide each other, so the minimal polynomials cannot divide each other... but obvsiouly, there is nothing saying that exists some algebric element, so what can I do?
Clearly $F=L\cap K$ is a field extension of $k$ such that both $L$ and $K$ are field extensions of $F$. Then $$m=[L:k]=[L:F][F:k]$$ and $$n=[K:k]=[K:F][F:k]$$ implies that $[F:k]$ is a common divisor of $m$ and $n$. What next?