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Generator (number theory)

  1. Oct 30, 2007 #1
    1. The problem statement, all variables and given/known data

    Let a be a generator of [tex]F_q[/tex]

    Prove that [tex]a^i[/tex] is a generator if & only if [tex]i[/tex] and [tex]q-1[/tex] are relatively prime.

    2. Relevant equations

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

    relatively prime means that [tex]gcd(i,q-1)[/tex]=1

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

    3. The attempt at a solution

    Suppose that [tex]a^i[/tex] is a generator of [tex]F_q[/tex]. then a^(i(q-1)) =1 (mod q)

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

    How does that sound?
  2. jcsd
  3. Oct 31, 2007 #2
    Think of the subgroup generated by a^i.
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