Can somebody help me out, I was thinking about a similar attempt.
If has a primitive root that means under addition and multiplication will be a finite field.
Because finite fields are power of primes.
Apparently, I missed something. I happen to know that I am wrong because it is a well known theorem that I heard of it before. Can someone spot an error?
EDIT: I spotted my error. I was assuming which is only true when it is a prime. I will now see if I can answer this question elegantly via finite fields.