I have a question regarding some claims I read about the Law of Quadratic Reciprocity that I can't fully understand. The law itself is written as follows:
For all odd numbers $P,Q \in \mathbb{N}$ with $\gcd(Q,P) = 1$ we have: $\left(\frac{Q}{P}\right) = (-1)^{(P-1)(Q-1)/4}\left(\frac{P}{Q}\right)$
The claims that I read are that from here it is implied that for an odd prime $p \in \mathbb{N}$ and $a \in \mathbb{Z}^{*}_{p}$ if $p \equiv 3\bmod\ 4$, either $a$ or $-a$ is a quadratic residue, while if $p \equiv 1\bmod 4$ both $a$ and $-a$ are either quadratic residues, or quadratic nonresidues.
Now I don't have much experience when it comes to this area of math, so these claims are not that obvious to me just by looking at the law of quadratic reciprocity, so I wanted to ask kindly if someone can perhaps shed some light onto why is this the case?
Hint: Given that you know the supplemental rules, you can compute \begin{equation*} \left(\frac{-a}{p}\right) = \left(\frac{-1}{p}\right)\left(\frac{a}{p}\right). \end{equation*} in each of the cases you are interested in.