# Primitive element modulo p

1. Dec 1, 2009

### hope2009

Suppose that p and q are odd primes and p=2q+1. Suppose that
α∈ Z_p^*,α≢±1 mod p.

Prove that α is primitive element modulo p if and only if α^q≡-1 mod q.

2. Dec 1, 2009

### robert Ihnot

That sounds like a homework problem?? Do you think?

3. Dec 2, 2009

### robert Ihnot

I don't get any response on this. But since q is a "Sofie Germain Prime," then q=(p-1)/2.

By Fermat's Little Theorem z^(p-1) ==1 Mod p. IF a number is a primitive root--which I take is what he means here- no extensions are mentioned; then all its powers Modulo p must differ, giving us the p-1 residue system. ;

To do this, a^(p-1)/2 ==-1 Mod p. Since if it was 1, then a would not generate all the elements.

Last edited: Dec 2, 2009
4. Dec 3, 2009

### hope2009

thanks alot i really appreciate your help

5. Dec 3, 2009

### robert Ihnot

hope2009: thanks alot i really appreciate your help

Happy to hear you are satisifed. However, there is an error I see now:

Prove that α is primitive element modulo p if and only if α^q≡-1 mod q.

You mean $$a^q \equiv -1 \bmod p$$ Since by Fermat's little theorem for a prime, $$a^q \equiv a \bmod q$$

Last edited: Dec 3, 2009