Quadratic Cyclotomic Polynomials

Question

Please can anyone help me out with the following: 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

If $p$ is 1 mod 3, then the order (i.e. $p-1$) of the multiplicative group $\mathbb{Z}_p^*$ is divisible by 3, hence this group has an element $\alpha$ of order 3, hence $\alpha^3 - 1 = 0$ but $\alpha - 1 \neq 0$, and hence $\alpha$ is a root of $(x^3-1)/(x-1) = x^2 + x + 1$.

If $p$ is 2 mod 3, then the order of $\mathbb{Z}_p^*$ is not divisible by 3, and hence $\mathbb{Z}_p^*$ has no element of order 3, and thus, no root of $x^2+x+1$.