Primitive root criterion

Question

Let $p$ be a prime, $g\in (\mathbb{Z}/ p\mathbb{Z})$* a primitive root.

I want to find a necessary and sufficient condition on $a\in \mathbb{Z}$ such that $g^a$ is also a primitive root.

My conjecture is that $gcd(a,p-1)=1$ is that condition. To test this, I first suppose that $gcd(a,p-1)=1$. If I get that $g^a$ is a primitive root, then my condition is sufficient. Then I suppose $gcd(a,p-1)=d$ where $d>1$ and in $\mathbb{N}$. If I get that $g^a$ is not a primitive root, then my condition is necessary.

First, I suppose $gcd(a,p-1)=1$ By the definition of a primitive root, we have $g^{p-1}\equiv 1\pmod p$ with $p-1$ the smallest integer for which the latter congruence is true. This implies $(g^{p-1})^a\equiv 1\pmod p$ and $(g^a)^{p-1}\equiv 1\pmod p$

If $p-1$ is the multiplicative order of $g^a$, we're done.

If $p-1$ is not the multiplicative order of $g^a$, then $\exists\space d\in \mathbb{N}$ such that $d|p-1$ and $\frac{p-1}{d}$ is the multiplicative order of $g^a$.

So we get $(g^a)^{\frac{p-1}{d}}\equiv 1\pmod p$ which implies $(g^{\frac{a}{d}})^{p-1}\equiv 1\pmod p$

I expected to arrive at some contradiction because $a$ and $p-1$ being coprime, it implies that $d$ does not divide $a$ but I'm not sure where the contradiction lies or if there is any. Any tips?

Answer 1

I see several mistakes there. Firstly, your argument for sufficient condition does not prove anything, both your statements about $(g^a)^{p-1}$ are true for any $a$. You want to prove $(p-1)$ is the smallest number for which that equality is true. And it is, because the smallest common multiple of $a$ and $p-1$ is $a \cdot (p-1)$, because $gcd (a,p-1)=1$.

Now, the reverse. It is similar, only now we know that the smallest common multiple of $a$ and $p-1$ is $\frac{a \cdot (p-1)}{gcd (a,p-1)}<a \cdot (p-1)$, therefore there is a number smaller than $a(p-1)$, for which we can get to $1$, because now $(g^a)^{\frac{p-1}{gcd (a,p-1)}}=1$ as well. Therefore it does not meet criteria of a primitive root.