How to give irreducible factorization in $\mathbb{Z}_5$?

Question

I have this polynomial:

$$f=x^5+2x^3+4x^2+x+4$$

How can i find the irreducible factorization(in $\mathbb{Z}_5$)?I can find the roots easily but thats not enough.

Answer 1

Since $\pm2$ are zeroes of the polynomial, $f$ has linear factors $x-2$ and $x+2$. We have $f(x) = (x-2)(x+2)(x^3+x-1)$. The quotient $x^3+x-1$ is cubic and has no linear factor over $\mathbb{F}_5$, hence is irreducible over $\mathbb{F}_5$.