Prove Proposition (7.6).
Let's remind ourselves of Proposition (7.6).
Let $a$ be any integer and $p$ an odd prime, then
$$ a^{\frac{p-1}{2}} \equiv \pm 1 \pmod p $$
provided $p \not \mid a$.
We start with Fermat's Little Theorem (4.1), applicable since $p \not \mid a$,
$$ a^{p-1} \equiv 1 \pmod p $$
Since $p-1$ is even, we have by Proposition (3.14)(b)
$$ a^{\frac{p-1}{2}} \equiv \pm 1 \pmod p $$