What are the primitive roots of Z_32? [itex]\varphi[/itex]([itex]\varphi(32)[/itex])=8 However you must first check that there is a primitive root. A PR exists if (a) n=2,4 (b) n=p^k (c)n=2p^k According to the solutions, Z_32 has no primitive roots. Is this correct? 32=2^5 which fulfills one of the conditions (b) so shouldn't it have PRs?