Is a^i a Generator of F_q If and Only If i and q-1 Are Relatively Prime?

[rad0786]

Homework Statement

Let a be a generator of $$F_q$$

Prove that $$a^i$$ is a generator if & only if $$i$$ and $$q-1$$ are relatively prime.

Homework Equations

a is a generator of $$F_q$$ means that a^(q-1) = 1 and $$a^i$$ cannot be 1 for all i not q-1.

relatively prime means that $$gcd(i,q-1)$$=1

fermats theorem says that: a^(p-1) = 1 (mod p ) where p is prime

The Attempt at a Solution

=>
Suppose that $$a^i$$ is a generator of $$F_q$$. then a^(i(q-1)) =1 (mod q)

so by fermats theorem, gcd(i, q-1) = 1?

How does that sound?

 

[zhentil]

Think of the subgroup generated by a^i.