Proof on the Legendre Symbol

Question

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?

Answer 1

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.