Let $p$ (be an odd prime) of the form $p \equiv 3 \pmod 4$. Also let $r$ be a primitive root modulo $p$. Prove that $-r$ has order $\frac{p-1}{2}$.
We proved this as a lemma in order to solve the last Exercise (6.4).9.
The only difference is that $p \equiv 3 \pmod 4$ means $p-1 = 2 + 4t$ for some integer $t$, that is $\frac{p-1}{2}=1+2t$, and so $\frac{p-1}{2}$ is odd, the assumption for the lemma we proved.