Decomposition of a prime number in a cyclotomic field Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th  root of unity in $\mathbb{C}$.
Let $A$ be the ring of algebraic integers in $\mathbb{Q}(\zeta)$.
Let $p$ be a prime number such that $p \neq l$.
Let $f$ be the smallest positive integer such that $p^f \equiv 1$ (mod $l$).
Then $pA = P_1...P_r$, where $P_i's$ are distinct prime ideals of $A$ and each $P_i$ has the degree $f$ and $r = (l - 1)/f$.
My question: How would you prove this?
This is a related question.

Once you know what the ring of integers in $\mathbf{Q}(\zeta)$ is, you know that the factorization of a rational prime $p$ therein is determined by the factorization of the minimal polynomial of $\zeta$ over $\mathbf{Q}$, which is the cyclotomic polynomial $\Phi_\ell$, mod $p$. So you basically just need to determine the degree of a splitting field over $\mathbf{F}_p[X]$ of the image of $\Phi_\ell$ in $\mathbf{F}_p$. The degree is the $f$ in your question. This can be determined using the Galois theory of finite fields, mainly the fact that the Galois group is cyclic with a canonical generator. The details are carried out in many books, e.g., Neukrich's book on algebraic number theory.
