$1+x+x^2 + \cdots + x^{p-1}$ in finite field

Question

In $\mathbb{Q}$, $1+x+x^2+\cdots+x^{p-1}$ is always irreducible by Eisenstein criterion.

What is an example of a finite field $\mathbb{F}_q$, and a prime $p$ such that $1+x+x^2+\cdots+x^{p-1}$ is not irreducible in $\mathbb{F}_q$?

Answer 1

The first examples that come to mind are:

• $p=7$, $q=2$. Modulo $2$ we have $$1+x+x^2+x^3+x^4+x^5+x^6=(1+x+x^3)(1+x^2+x^3).$$
• $p=3$, $q=7$. Modulo $7$ we have $$1+x+x^2=(x-2)(x-4).$$

Earlier today I explained how to figure out when a cyclotomic polynomial $\Phi_n(x)$ is irreducible over a prime field $\Bbb{F}_q$. This is a special case of that general result.

Answer 2

For $q=p=3$, consider $1+x+x^2=(x+2)^2$. Hence reducible.