Irreducibility over $\mathbb{F}_5$

Question

I am trying to prove (or disprove) that the polynomial $\sum_{i=0}^{30} x^i$ is irreducible over $\mathbb{F}_5$.

I have a guess that it is enough to prove that any primitive 31th root of the unit belongs to that field but I dont know why this is true.

I will be thankful for any hint or discussion!

Have a great sunday!

Answer 1

This polynomial is $(x^{31}-1)/(x-1)$ which is the $31$'st cyclotomic polynomial. It splits over $\mathbb{F}_{5^3}$ because 31 divides $5^3-1$. This means that all irreducible factors over $\mathbb{F}_5$ will have degree 3 (degree 1 factors do not occur since 31 does not divide $5-1$ (= the order of the multiplicative group $\mathbb{F}_5^*$)).