Factorising polynomial into irreducible ones.

Question

The task is to factorise following polynomial into irreducible ones over ring $\mathbb{Z_5}[x]$:
$$f=x^5+3x^4+x^3+x^2+3.$$

While I've solved similar task with simple field $\mathbb{Z}$, have no idea how to deal with it in case of $\mathbb{Z_5}$.

Answer 1

Hint $\,\ \overbrace{x\color{#0a0}{x^4}\!+\!3\color{#0a0}{x^4}\!+\!x^3\!+\!x^2\!+\!3}^{\textstyle\! x\ \:\!+\:\!\ 3 \ \:\!+\:\!\ x^3\!+\!x^2\!+\!3} \:\!\bmod\, \color{#0a0}{x^4-1} = \dfrac{x^4-1}{x-\color{#c00}1} $ so $\,f\,$ has roots $\,\overline{\{\color{#90f}0,\color{#c00}1\}} = \{2,3,4\}$

(recall by Fermat every $\,a\in\Bbb Z_5\,$ is a root of $\,x^5 -x,\,$ hence every $\,a\neq\color{#90f} 0$ is a root of $\,x^4-1).$

Remark $ $ This method is essentially a special case of very general methods for irreducibility and factorization of polynomials over finite fields. Follow the link for examples of similar problems solved this way.