2
$\begingroup$

Let $p$ be a prime, then is it true that a generator $g$ of $\mathbb{Z}^*_p$ cannot be a quadratic residue modulo $p$? And if yes why?

$\endgroup$
  • 2
    $\begingroup$ Think about it: If $g$ is a quadratic residue then so is any power of $g$. (P.S. $p=2$ is different. You need to consider only odd primes.) $\endgroup$ – Harald Hanche-Olsen Apr 29 '16 at 13:58
6
$\begingroup$

Yes, it is true, for odd $p$.

The point is that a generator $g$ has order $p-1$, which is even.
Yet, if $g= a^2$, then $g^{(p-1)/2} = a^{p-1} = 1$, a contradiction to $p-1$ being the order of $g$.

$\endgroup$
  • 1
    $\begingroup$ Very nicely put and amazingly simple. +1 $\endgroup$ – DonAntonio Apr 29 '16 at 20:05

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.