Integer to Power of p-1 over 2 Modulo p

Theorem
Let $$a \in \Z$$, and let $$p$$ be an odd prime.

Let $$\left({\frac{a}{p}}\right) = a^{\frac{(p-1)}{2}} \pmod p$$ be the Legendre symbol.

Then:

\left({\frac{a}{p}}\right) = \begin{cases} 0 & : a \,\bmod\, p = 0 \\ +1 & : \exists x \in \Z: x^2 \equiv a \pmod{p} \\ -1 & : \text {otherwise} \end{cases}$$

Proof
Let $$b = a^{\frac{(p-1)}{2}}$$.

We have that $$b^2 = a^{p - 1}$$.

If $$b^2$$ divides $$p$$, then $$b \,\bmod\, p = 0$$.

Otherwise, from Fermat's Little Theorem, $$b^2 \equiv 1 \pmod p$$.

That is, $$b^2 - 1 \equiv 0 \pmod p$$.

Consider $$b^2 - 1 = \left({b+1}\right) \left({b-1}\right)$$ from Difference of Two Squares.

So either $$b+1$$ divides $$p$$ or $$b-1$$ does.

(They can't both do, as they have a difference of $$2$$, and $$p$$ is an odd prime.

From Congruence Modulo Negative Number‎, we have that $$p-1 \equiv -1 \pmod p$$.

Hence the result.