factorization of the cyclotomic $\Phi_n(x)$ over $\Bbb F_p$

One can prove that the $\Phi_n(x)$ are irreducible over $\Bbb Z$. Where $\Phi_n(x)=\prod _{(a,n)=1}\zeta_n^a$ (i.e the product of the primitive n-rooth of unity). I want to find a factorization of $\Phi_n$ in $\Bbb F_p$. I proved that all the irreducible factors of $\Phi_n$ over $\Bbb F_p$ are of the same degree, and the degree of all of them is $d$. Where $d$ is the order of $p \in \Bbb Z_n^*$. And since $ Degree (\Phi_n ) = \phi(n) $, then $\Phi_n$ is a product of $ \frac{\phi(n)}{d}$ irreducible factors of degree $d$. To find an explicit factorization , one way is to put a system of equations to find the coefficients of the irreducible factors, but I think that it’s very complicated.

I want to know if there is one way to do this, maybe considering what I said but more. Maybe some relation in the irreducible factors, would reduce the system or something like that.

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Those factorisations follow the factorisation of $\Phi_n$ on intermediate fields between $\mathbb Q$ and $\mathbb Q(\zeta_n)$.

For example, let $n=12$. There are $4$ cases :

If $p = 1 \pmod {12}$, then $\Phi_{12}$ splits in $\mathbb F_p$, so you have to find one primitive root of $1$, $\zeta_{12}$, then write $\Phi_{12}(X) = (X- \zeta_{12})(X- \zeta_{12}^5)(X- \zeta_{12}^7)(X- \zeta_{12}^{11})$

If $p = 5 \pmod {12}$, then it has the same factorisation as the one on $\mathbb Q(i)$, which is $(X^2-iX-1)(X^2+iX-1)$, so you only need to find one of the two square roots of $ -1$ in $\mathbb F_p$ to get the factorisation.

If $p = 7 \pmod {12}$, then it has the same factorisation as the one on $\mathbb Q(j)$, which is $(X^2-j)(X^2-j^2)$, so you only need to find one of the two primitive third roots of $1$ in $\mathbb F_p$ to get the factorisation.

If $p = 11 \pmod {12}$, then it has the same factorisation as the one on $\mathbb Q(\sqrt 3)$, which is $(X^2-\sqrt 3 X + 1)(X^2+ \sqrt 3 X + 1)$, so you only need to find one of the two square roots of $3$ in $\mathbb F_p$ to get the factorisation.

In any case, finding those factorisations mean that you have to solve some algebraic equations in $\mathbb F_p$ (they are not always simple roots for example if $n=7$ you may have to find $\cos(2\pi/7)$ which is a root of a degree $3$ equation while it’s not a cube root), and in some cases, you have to find one primitive $n$th root of $1$ root in $\mathbb F_p$.

I don’t think there’s a big leap in difficulty in simply finding one primitive root $\zeta_n$ in $\mathbb F_{p^d}$ and then computing the irreducible factors which are all the $(X- \zeta_n^k)(X – \zeta_n^{kp}) (X – \zeta_n^{kp^2})\ldots (X – \zeta_n^{kp^{d-1}})$