I haven't fully wrapped my head around primitive roots yet and I have a question with them:
Let $p$ be an odd prime and $g$, $h$ be two primitive roots modulo $p$. Show that $gh$ is not a primitive root modulo $p$.
I think I'll need to use the fact that if $g$ is a primitive root modulo p then a reduced residue system modulo $p$ is $g$, $g^2$,…, $g^{p-1}$
Any help would be much appreciated!
Best Answer
Hint:
$$g^{(p-1)/2}\equiv h^{(p-1)/2}\equiv-1\pmod p$$
$$(gh)^{(p-1)/2}\equiv?$$