Let $n \geq 1$ be an integer, let $\zeta$ be a primitive $n$th root of unity, and let $P$ be a prime ideal of the ring of integers of $\mathbf{Q}(\zeta)$.
What is the multiplicative order of $\zeta$ modulo $P$?
By multiplicative order I mean the least integer $k \geq 1$ such that $\zeta^k \equiv 1 \bmod P$. Let's call it $\operatorname{ord}_P(\zeta)$. I am particularly interested in the case in which $\operatorname{ord}_P(\zeta) = n$ so that $\zeta$ is somehow a primitive $n$th root of unity "modulo $P$".
It is easy to see that $\operatorname{ord}_P(\zeta)$ divides $\gcd(n, N(P) - 1)$, but I could not find more information.
Thank
"Usually" it's $n$. It can only be less than $n$ if $P$ contains an element $\zeta^m-1$ where $m$ is a proper factor of $n$. That only occurs when $P\cap\Bbb Z=p\Bbb Z$ where $p$ is a factor of $n$, so only at the finitely many ideals "over" a prime factor of $n$.