Lang's Chapter 6, Corollary 3.2 states that
if $m,n$ are relatively prime, integers greater than $1$, then $$\mathbb{Q}(w_m) \cap \mathbb{Q}(w_n)=\mathbb{Q}$$ where $w_m$ is the primitive $mth$ root of unity.
Again the proof is not clear. It states that $ \mathbb{Q}(w_m) \mathbb{Q}(w_n)=\mathbb{Q}(w_{mn})$ and says it follows from the multiplicatively of Euler Totient Function. I couldn't understand how it follows immediately.
What I decoded is the following.
(1)-I will show that $ \mathbb{Q}(w_m) \mathbb{Q}(w_n)=\mathbb{Q}(w_{mn})$
Proof of (1)-Its trivial that $\mathbb{Q}(w_{mn})$ contains both $\mathbb{Q}(w_{m})$ and $\mathbb{Q}(w_{n})$
On the other hand, it is sufficent to show that $\mathbb{Q}(w_{m})\mathbb{Q}(w_{n})$ contains a primitive root of unity. Infact $w_m w_n$ works for us. When $(m,n)=1$, we do have this result. It can be seen by writting the primitive root of unity as a power of $e$.
But how does it imply that
$$\mathbb{Q}(w_m) \cap \mathbb{Q}(w_n)=\mathbb{Q}$$
