Irreducible Quadratic Polynomial over $\mathbb{F}_p$ when $p \equiv 1 \pmod 4$

Question

It is well known that the polynomial $x^2+1$ is irreducible over $\mathbb{F}_p$ if and only if $p \equiv 3 \pmod 4$. I was wondering if there is another universal degree $2$ polynomial that is irreducible over $\mathbb{F}_p$ if and only if $p \equiv 1 \pmod 4$.

Answer 1

For $p\ne 2$ the factorization of $X^2+bX+c\bmod p$ is given by $(\frac{b^2-4c}{p})$, quadratic reciprocity says $(\frac{b^2-4c}{p})= f(p)$ with $f$ completely multiplicative and periodic of least period $\Delta$, in your assumption there is that $f(5)=f(13)=-1$ which implies that $f(65)=1$ and $\gcd(65,\Delta)=1$, thus $f(p)=1$ whenever $p\equiv 65 \bmod \Delta$, with Dirichlet theorem on arithmetic progressions take a prime $p\equiv 1 \bmod 4,p\equiv 65 \bmod \Delta$ to get $f(p)=1$.