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.