Equivalence of Definitions of Legendre Symbol

Theorem
Let $p$ be an odd prime.

Let $a \in \Z$.

Proof
From Integer to Power of $\dfrac {p - 1} 2$ Modulo $p$, one of the following cases holds:
 * $(1): \quad a^{\frac{(p - 1)} 2} \bmod p = 0$
 * $(2): \quad a^{\frac{(p - 1)} 2} \bmod p = 1$
 * $(3): \quad a^{\frac{(p - 1)} 2} \bmod p = p - 1$

Let $\left({\dfrac a p}\right)$ be defined as the Legendre symbol by definition 1:

By Congruence Modulo Negative Number we have:
 * $p - 1 \pmod p = - 1$

Thus $\left({\dfrac a p}\right)$ is the Legendre symbol by definition 2: