Different formulations of the Law of Quadratic Reciprocity

Question

The law of quadratic reciprocity is given as:

$(\frac{p}{q})(\frac{q}{p}) = (-1)^{((p-1)/2)((q-1)/2)}$

Apparently we can say:

$(\frac{p}{q}) = (\frac{q}{p})(-1)^{((p-1)/2)((q-1)/2)}$

and

$(\frac{q}{p}) = (\frac{p}{q})(-1)^{((p-1)/2)((q-1)/2)}$

But I'm not really sure how we arrive at those conclusions. Could anyone help me out?

Answer 1

$$\left(\frac{p}{q}\right)\left(\frac{q}{p}\right) = (-1)^{((p-1)/2)((q-1)/2)}$$

multiplying both sides by $\color{blue}{\left(\frac{q}{p}\right)}$, we get $$\left(\frac{p}{q}\right)\left(\frac{q}{p}\right)\color{blue}{\left(\frac{q}{p}\right)} = \color{blue}{\left(\frac{q}{p}\right)}(-1)^{((p-1)/2)((q-1)/2)}$$

Next notice that $\left(\frac{q}{p}\right)\color{blue}{\left(\frac{q}{p}\right) } = \left(\frac{q^2}{p}\right) = 1$ to get the desired result $$\left(\frac{p}{q}\right) = \color{blue}{\left(\frac{q}{p}\right)}(-1)^{((p-1)/2)((q-1)/2)}$$

Answer 2

Remember that Legendre symbols are always $\pm 1$, and so square to $1$.