Prove that the Legendre symbol
$$ (\frac{−1}{p}) = (−1)^{\frac{p−1}{2}} $$
where $p$ is an odd prime.
We will use Proposition (7.8).
Let $p$ be an odd prime and $a$ be an integer such that $p \not \mid a$. Then
$$(\frac{a}{p} ) \equiv a^{\frac{p−1}{2}} \pmod p$$
Setting $a=-1$ gives us the desired result.
$$(\frac{-1}{p} ) \equiv (-1)^{\frac{p−1}{2}} \pmod p$$