Finding Factors of $x^4+x^3+x^2+x+1 \in \mathbb{Z}_5[x]$

Question

For $f(x)=x^4+x^3+x^2+x+1 \in \mathbb{Z}_5[x]$, I have found that $1$ is a zero of this polynomial, so by the factor theorem I know that for some $q(x) \in \mathbb{Z}_5[x]$ I should have $f(x)=(x-1)q(x)$. My trouble is, when I try the division algorithm I will always get a remainder of $1$.

How can I look for an explicit factorization of this polynomial?

Answer 1

$$x^4+x^3+x^2+x+1=\frac{x^5-1}{x-1}$$ But modulo $5$, $x^5-1\equiv (x-1)^5$. Therefore over the field of five elements, $$x^4+x^3+x^2+x+1=(x-1)^4.$$

Answer 2

You are free to change the polynomial in a cosmetic manner, by adding or subtracting multiples of 5 from the coefficients. For example, $$ x^4 + x^3 + x^2 + x + 1 \; \; \equiv \; \; \; x^4 - 4 x^3 + 6 x^2 - 4 x + 1 $$ where you might recognize some binomial coefficients.