question is as follows:
Let $p$ be a prime with $p \equiv 1 \mod 4$, and $r$ be a primitive root of $p$. Prove that $-r$ is also a primitive root of $p$.
I have shown that $-r^{\phi(p)} \equiv 1 \mod p$. What I am having trouble showing, however, is that the order of $-r$ modulo $p$ is not some number (dividing $\phi(p)$ that is LESS than ($\phi(p)$).
By way of contradiction, I've shown that the order of $-r$ cannot be an EVEN number less than $\phi(p)$. But my methodology does not work for the hypothetical possibility of an ODD order that is less than $\phi(p)$.
Any and all help appreciated. Happy to show methodology for any of the parts I have managed to do, if requested.