I don't understand the latter half of the proof of J. Neukirch "Algebraic Number Theory" chapter 1 section 10 proposition 10.3.
The situation is below:
Let $n$ be a natural number, $p$ be a prime not dividing $n$, and $\mathfrak{p}$ be a prime ideal dividing $p$.
And let $\zeta$ be a primitive $n$-th root of unity, $\phi_n(X)$ be a minimal polynomial of $\zeta$.
First, let $\mu_n(R)$ be a set of $n$-th root of unity in a ring $R$, then a natural homomorphism $\mathcal{o}\rightarrow \mathcal{o}/\mathfrak{p}\ (\mathcal{o}\colon =\mathbb{Z}(\zeta)$) induces "bijection"*1 $\mu_n(\mathcal{o})\rightarrow\mu_n(\mathcal{o}/\mathfrak{p})$ and "this sends primitive to primitive" *2.
Next, let $f_p$ be a minimum number with $p^{f_p}\equiv 1\ {\rm mod}\ n$, and we have "$\mathbb{F}_{p^{f_p}}$ is splitting field of $\bar{\phi}_n(X)\colon =\phi_n(X)\ {\rm mod}\ p$".*3
Finally, considering factorization of $\bar{\phi}_n(X)$ on $\mathbb{F}_p$, "all factors are the minimal polynomial of primitive $n$-th root of unity $\bar{\xi}$, and so their degree are $f_p$".*4
What I don't understand are descriptions which are sandwiched by " "(four questions). As for the first "bijection", it is enough the reason of only surjectivity.
And more, I want ask people who have this book where is the definition of $f_p$ used when it is clear we can take $n$ being not divided by $p$(at the middle of this proof).
I'll appreciate even if partial answers.