Possible Duplicate:
Irreducible factors of $X^p-1$ in $(\mathbb{Z}/q \mathbb{Z})\[X\]$
Let $p$ be a prime number. Let $l$ be an odd prime number such that $l \neq p$.
Let $X^l - 1 \in \mathbb{Z}[X]$. Since $(X^l - 1)' = lX^{l-1}$, $X^l - 1$ has no multiple irreducible factor mod $p$. Since $X^l - 1 = (X - 1)(1 + X + ... + X^{l-1})$, $1 + X + ... + X^{l-1}$ has no multiple irreducible factor mod $p$, either.
Let $1 + X + ... + X^{l-1} \equiv f_1(X)...f_r(X)$ (mod $p$), where $f_i(X)$ is a monic irreducible polynomial mod $p$.
Let $f$ be the smallest positive integer such that $p^f \equiv 1$ (mod $l$).
My question: Can we prove that the degree of each $f_i(X)$ is $f$?