I am learning about cyclotomic fields (hoping to understand the basics of Iwasawa theory) using Washington, Introduction to Cyclotomic Fields. I was reading this proposition, it seems to be using "quite advanced" results such as
If $K$ is a number field and $d_K$ is the discriminant of $K$, then
$$ p|d_K \text{ if and only if } p \text{ ramifies in } K$$
and $$ |d_K| > 1 \text{ if } K \neq \mathbb{Q}$$
which follows from Minkowski's bound according to Wikipedia.
According to Lang (the bottom image),
which somehow magically proves the same statement from
$$ \mathbb{Q}(\zeta_n) \mathbb{Q}(\zeta_m) = \mathbb{Q}(\zeta_{mn}) $$
Am I missing something, or is it just a typo? Thank you in advance.
Here is what I think Lang had in mind: earlier in the same chapter, Lang proves that if $F$ and $K$ are extensions of a field $k$ with $K$ Galois over $k$, then $KF$ is Galois over $F$ and $K$ is Galois over $K\cap F$, with $\mathrm{Gal}(KF/F)\simeq \mathrm{Gal}(K/K\cap F)$. In particular, $$ [KF:F]=[K:K\cap F]$$
In your problem, let $K=\mathbb{Q}(\zeta_m)$, $F=\mathbb{Q}(\zeta_n)$, and $k=\mathbb{Q}$. Then $KF=\mathbb{Q}(\zeta_{mn})$, hence the above equality becomes $$ [\mathbb{Q}(\zeta_{mn}):\mathbb{Q}(\zeta_n)]=[\mathbb{Q}(\zeta_m):\mathbb{Q}(\zeta_m)\cap\mathbb{Q}(\zeta_n)]$$ Now the left-hand side is equal to $$ \frac{[\mathbb{Q}(\zeta_{mn}):\mathbb{Q}]}{[\mathbb{Q}(\zeta_{n}):\mathbb{Q}]}=\frac{\phi(mn)}{\phi(n)}=\phi(m)$$ by the tower rule, hence the right-hand side must also be $\phi(m)=[\mathbb{Q}(\zeta_m):\mathbb{Q}]$. But this implies that $\mathbb{Q}(\zeta_m)\cap\mathbb{Q}(\zeta_n)=\mathbb{Q}$.