Proof on the Legendre Symbol I'm working on an exercise involving the Legendre Symbol. It gives me a hint but I'm not sure how to prove it.
Let p and q be odd prime numbers with $p = q + 4a$ for some $a \in \mathbb{Z}$. Prove that $(\frac{a}{p}) = (\frac{q}{p})$ 
Hint: Prove that $(\frac{a}{p}) = (\frac{-q}{p})$
I'm not really sure how to prove that the hint is even true. Could someone help me see how the hint is true?

Since $(4\,|\,p) = (2\,|\,p)^2 = 1$, 
$$\left(\frac{a}{p}\right) = \left(\frac{4}{p}\right) \left(\frac{a}{p}\right) = \left(\frac{4a}{p}\right) = \left(\frac{p-q}{p}\right).$$ Since $p - q\equiv -q\pmod{p}$, 
$$\left(\frac{p - q}{p}\right) = \left(\frac{-q}{p}\right).$$
This establishes the hint.
