Legendre Symbol: show that $\left(\frac{-3}p\right)=\left(\frac p3\right)$

Question

I am asked to solve the following question

Let $p$ be a prime, $p>3$. By considering the cases $p ≡1\bmod4$ and $p≡-1\bmod4$ separately, show that $\left(\frac{-3}p\right)=\left(\frac p3\right)$.

So if $p ≡ 1\bmod4$, $\left(\frac{-3}p\right)=\left(\frac p{-3}\right)$ and if $p ≡ -1\bmod4$, $\left(\frac{-3}p\right)=-\left(\frac p{-3}\right)$.

And then I get stuck on how to process on; is there anything I can do to deal with the minus sign of $3$?

Thanks

Answer 1

Hint: Use the fact that $-1$ is a quadratic residue mod $p$ if and only if $p\not\equiv3\pmod{4}$.