Field Extension equivalence

Question

Let $K ⊆ L$ be an algebraic field extension and $u, v ∈ L$ with $[K(u) : K] = m$ and $[K(v) : K] = n$. Show that $[K(v)(u) : K(v)] = m \iff [K(u)(v) : K(u)] = n$, and both equalities hold if $m$ and $n$ are coprime.

Answer 1

As you've mentioned you should use the Tower Law. You would get:

$$[K(v)(u):K] = [K(v)(u):K(v)][K(v):K] = [K(v)(u):K(v)]n$$

Also:

$$[K(v)(u):K] = [K(u)(v):K(u)][K(u):K] = [K(u)(v):K(u)]m$$

Hence:

$$[K(u)(v):K(u)]\cdot m = [K(v)(u):K(v)]\cdot n$$

From here you should be able to derive the conclusion