Irreducibility of polynomial in $\Bbb F_p[x]$

Question

What can we say about the irreduciblity of $x^{q-1} + \cdots + 1 $ in $\Bbb F_p[x]$ where $p,q$ distinct primes?

In $\Bbb Z[x]$ we may apply a transformation and apply Eisenstein's criterion. But $\Bbb F_p$ has no more prime ideals.

Answer 1

$x^{q-1}+x^{q-2}+\cdots+x+1$ is the $q$-th cyclotomic polynomial. It is irreducible over $\Bbb F_p$ iff $p$ is a primitive root modulo $q$. This is because adjoining the $q$-th roots is unity to $\Bbb F_p$ gives $\Bbb F_{p^k}$ where $k$ is the least integer with $q\mid (p^k-1)$.