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

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.

• When $x^3 = 1$ has a solution (other than $1$) in $\mathbb{F}_p[X]$? Sep 5 '21 at 7:16
• Consider what you can say about the existence of an element of order 3 in the group $F_p^*$. Sep 5 '21 at 7:22
• Multiply your polynomial by $X-1$. When does that product have three roots in $\Bbb F_p$? Sep 5 '21 at 7:39

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 $$\{r, r^2, \dots, r^{p-2}\} = \{2, 3, \dots, p-1\} = \mathbf{F}_p \setminus \{0,1\}$$?

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 $$\left(\frac{-3}{p}\right)$$.

• Can you please tell how it reduces to quadratic residues of -3? I am not able to get it Sep 18 '21 at 12:58
• I have studied quadratic residues from David Burton's elementary number theory but I am unable to find a result there in terms of $p\equiv 1 (mod 3)$ . The results there are in form of $p\equiv (mod 4)$ and $p\equiv (mod 12)$. So, a detailed answer will very much help me. Sep 18 '21 at 13:08
• $X^2+X+1$ is reducible $\iff (2X+1)^2 + 3=0$ has a root $\iff -3 \equiv (2X+1)^2 \iff -3$ is a square mod $p$. Oct 28 '21 at 0:36