How to find irreducible factors of the polynomial $p(x) = x^5 -1$ over integers modulo $5$

Question

Ok so I'm trying to answer this question. I've started by using the factor theorem to determine that $\ p(x) = x^5 -1 =(x-1)(x^4 + x^3 +x^2 +x +1)\ $ I have no idea what to do next to reduce this polynomial over the modulo 5 field. If someone could provide me with the next steps in an algorithmic form, that would be excellent. Thank you.

Answer 1

As a polynomial in $\mathbb{Z}_5[x]$, $x^5-1=(x-1)^5$.