Quadratic Cyclotomic Polynomials

Question

Consider the quadratic cyclotomic polynomial $\phi$$(x)$ $=$ $x$$^2$$+1$. Let $p$ be a prime number. Want to prove that $\phi$$(x)$ is irreducible in $\Bbb Z$$_p$$[x]$ if and only $p$ $\equiv$ $3$ mod $4$.

Please can anyone help me out here?

Answer 1

$\phi(x)$ is reducible in $\mathbb Z_p [x]$ iff $-1$ is a quadratic residue in $\mathbb Z_p$. The multiplicative group of units, $\mathbb Z_p^\times$, is cyclic. Using $g$ to denote a generator of $\mathbb Z_p^\times$, it is clear that $g^n$ is a quadratic residue iff $n$ is even. Since the order of $\mathbb Z_p^\times$ is $p-1$, we have $g^{(p-1)/2} = -1$. So $-1$ is a quadratic residue iff $(p-1)/2$ is even, which holds iff $p \equiv 1$ mod $4$.