Irreducible Quadratic Cyclotomic Polynomial

Question

Can anyone please help me out here?

Consider the quadratic cyclotomic polynomial $\phi$$(x)$ $=$ $x$$^2$$-x$$+1$. Let $p$ be a prime number. Want to prove that $\phi$$(x)$ is irreducible in $\Bbb Z$$_p$$[x]$ if and only $p$ $\equiv$ $2$ mod $3$.

Answer 1

For $p=2$ the polynomial $\phi(x)$ has no root in $\mathbb{F}_2[x]$ so it is irreducible. For $p=3$ we have $\phi(x) \equiv (x+1)^2 \pmod 3$. For prime $p>3$ a root of $\phi(x)$ is a primitive $6$-th root of unity and vice versa (since $x^6-1$ has no double root). Therefore for $p>3$ the polynomial $\phi(x)$ has a root in $\mathbb{F}_p[x]$ if and only if $p\equiv 1 \pmod 6$. Moreover $p \equiv 1 \pmod 6$ if and only if $p \equiv 1 \pmod 3$. Putting all these pieces together proves the given statement.