$X^2 +X +1$ is reducible in $\mathbb{F}_p [X]$ iff $p\equiv 1$ (mod 3)

Question

Edit : My question has been linked with following question and was marked as duplicate:Find all primes $p>2$ for which $x^2+x+1$ is irreducible in $\mathbb{F}_p[x]$ which is further linked with this question : $\mathbb{F}_p[X]/(X^2+X+1)$ is a field iff $p \equiv 2 \bmod 3$

In answer of 1 st linked question no answer is close to useful and in 2nd linked question is not exact duplicate but it doesnot answers my question completely and I can't ask the user: Zev Conoles because he is away for a long time.

So, I request you to reopen this question.

Answer of user Zev answers 1 side(assuming $X^2 +X+1 $ be reducible how to deduce that $p\equiv1 $(mod 3) but I have questions in that too: $X^2+X+1$ is reducible implies that $\mathbb{F}_p$ has a non trivial cube root of unity but how can I deduce $p\equiv 1 (mod 3)$ using that .

Also, it doesn't answers the converse that $p\equiv 2 $ (mod 3) implies that it is irreducible. So, please help with that.

This particular question was asked in a masters exam for which I am preparing.

Let p>3 be a prime number and $\mathbb{F}_p$ denote the finite field of the order p. Prove that the polynomial $X^2 +X+1$ is reducible in $\mathbb{F}_p [X]$ iff $p\equiv 1 $(mod 3).

I am really sorry but I will not be able to provide hint for any of the parts because I was unable to solve any of it.

I have done a graduate level course on Abstract Algebra but I am not able to solve it.

Kindly just tell what results to use . Rest I would like to work by myself.

Answer 1

Hint.

Let $r$ be a primitive root mod $p$. For $p \equiv 1 \pmod 3$, consider $r^{(p-1)/3} \in \mathbb{F}_p$. Is it a root of $x^2+x+1$?

Conversely if $p\not \equiv 1 \pmod 3$, $x=1$ is a root of $x^{3 }-1$. Are there any other roots among $\{r, r^2, \dots, r^{p-2}\} = \{2, 3, \dots, p-1\} = \mathbf{F}_p \setminus \{0,1\}$?

Answer 2

Hint:

$X^2+X+1$ is irreducible $\iff 4X^2+4X+4\equiv_p (2X+1)^2+3$ is irreducible. This reduces the problem to study $\left(\frac{-3}{p}\right)$.