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


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*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.


By the multiplicity property of the Legendre symbol, $(\frac{-3}{p})=(\frac{-1}{p})*(\frac{3}{p})$. 
Now $(\frac{-1}{p})=1$ if $p=1\mod 4$ and $(\frac{-1}{p})=-1$ if $p= 3 \mod 4$
Further, $(\frac{3}{p})=1$ if $p=1 $ or $ 11 \mod12$ and $(\frac{3}{p})=-1$ if $p=5$ or $ 7\mod 12$.
