For what odd prime p is -3 a quadratic residues? Non-residue? For what odd prime p is -3 a quadratic residues? Non-residue?


*

*Having a bit of trouble with this question, we are currently covering a section  on quadratic reciprocity and didn't really see anything in my notes that helped me solve this. Any help is greatly appreciated.


We have $-3$ is a QR of the odd prime $p\ne 3$ in two situations:
(i) $-1$ is a QR of $p$ and $3$ is a QR of $p$ or
(ii) $-1$ is a NR of $p$ and $3$ is a NR of $p$.
Case (i): $-1$ is a QR of $p$ if $p\equiv 1\pmod{4}$. Since $p\equiv 1\pmod{4}$, by Quadratic Reciprocity we have that the Legendre symbol $(3/p)$ is equal to $(p/3)$. Note that $(p/3)=1$ if $p\equiv 1\pmod{3}$, and $(p/3)=-1$ if $p\equiv -1\pmod{3}$.
So Case (i) occurs if $p\equiv 1\pmod{4}$ and $p\equiv 1\pmod{3}$, or equivalently if $p\equiv 1\pmod{12}$.
Case (ii): We must have $p\equiv 3\pmod{4}$. In this case, Reciprocity says that since $p$ and $3$ are both of the form $4k+3$, we have $(3/p)=-(p/3)$. Thus $(3/p)=-1$ if $p\equiv 1\pmod{3}$. 
So Case (ii) happens if $p\equiv 3\pmod{4}$ and $p\equiv 1\pmod{3}$, or equivalently if $p\equiv 7\pmod{12}$.
We can put the two cases together, and say that $-3$ is a quadratic residue of the odd prime $p$ if $p\equiv 1\pmod{6}$.
Thus $-3$ is a quadratic non-residue of the odd prime $p$ if $p\equiv 5\pmod{6}$.
