Understanding Neukirch´s proof

I´m studying algebraic number theory from Neukirch´s book. I´m reading the Proposition 10.2 which says:

A $$\mathbb{Z}$$-basis of the ring $$O$$ of integers of $$\mathbb{Q}(\zeta)$$ is given by $$1, \zeta, \zeta^2,..., \zeta^{d-1}$$, with $$d=\varphi(n)$$, in other words, $$O=\mathbb{Z}[\zeta]$$.

In one step of the proof, the author mentions, if $$n=l_1^{\nu_1}...l_r^{\nu_r}$$ and $$\zeta_i=\zeta^{n/l_i^{\nu_i}}$$ is a primitive $$l_i^{\nu_i}$$-th root of unity, one has

$$\mathbb{Q}(\zeta)=\mathbb{Q}(\zeta_1)...\mathbb{Q}(\zeta_r)$$, with $$\mathbb{Q}(\zeta_1)...\mathbb{Q}(\zeta_{i-1}) \cap \mathbb{Q}(\zeta_i)=\mathbb{Q}$$.

This is the step I cannot see. Could anyone explain me why these equalities holds? I''ll really appreciate this help!

• I think one part of the argument is that you should write $1$ as a linear combination with integer coefficients of the $n/l_i^{\nu_i}$. Oct 4, 2021 at 22:25

For cyclotomic fields in general we have $$\mathbb{Q}_a\mathbb{Q}_b=\mathbb{Q}_{\mathrm{lcm}(a,b)}$$ and $$\mathbb{Q}_a\cap\mathbb{Q}_b=\mathbb{Q}_{\gcd(a,b)}$$. The first statement is fairly elementary, while the second is explained here for instance. For more than two fields, you use induction.