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.