Ive been working through David Cox's book primes of the form $x^2+ny^2$, im stuck on problem 5.13, pg 118 and was wondering if i could get some hints, the problem is as follows:
K is a number field that contains an nth root of unity $\zeta$, let $O_{K}$ be the ring of integers of K. let $a \in O_{K}$ and let $p$ be a prime ideal in $O_{K}$ such that $na \not \in p$
a) prove that $1,\zeta,\dots,\zeta^{n-1}$ are distinct roots modulo $p$. The hint here is it show that $x^n-1$ is separable modulo $p$
b) use a) to prove $n | N(p)-1$ (here N(p) is Norm of $p$)
c) show that $a^{(N(p)-1)/n}$ is congruent to a unique root of unity modulo $p$.
There are more parts but this is where i need help. For part a) i wanted to use the fact that is gcd(f(x),f'(x))=1 then its separable, but im not sure how this works modulo $p$, or does it not really make a difference?
For part b) im not quite sure where the $N(p)-1$ comes from.
Thank you very much