# Primitive root

## Homework Statement

p prime, If p=1 ( mod 3) then Zp contains primitive cube roots of unity. Now I am considering which p does Zp contains primitive fourth roots of unity.

opposite way? I mean if p=1(mod4) then Zp contains primitive fourth roots of unity??

2. The attempt at a solution
I can prove that if Zp contains primitive fourth roots of unity, then 4|(p-1) . but how about the opposite way? I mean if p=1(mod4) then Zp contains primitive fourth roots of unity?? I know this statements true if q prime instead of 4. And what values of p does Zp contains primitive fourth roots of unity???

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## Homework Statement

p prime, If p=1 ( mod 3) then Zp contains primitive cube roots of unity. Now I am considering which p does Zp contains primitive fourth roots of unity.

opposite way? I mean if p=1(mod4) then Zp contains primitive fourth roots of unity??

2. The attempt at a solution
I can prove that if Zp contains primitive fourth roots of unity, then 4|(p-1) . but how about the opposite way? I mean if p=1(mod4) then Zp contains primitive fourth roots of unity?? I know this statements true if q prime instead of 4. And what values of p does Zp contains primitive fourth roots of unity???

Hi Funky1981!

The expression ##p \equiv 1 \pmod 3## means that there is a k such that ##p=3k+1##.

Now suppose g is a primitive root mod p.
Then ##g^{\phi(p)} \equiv g^{3k} \equiv 1 \pmod p##.
Therefore ##g^k## is a cube root of 1 in ##\mathbb Z_p##.

Same argument holds for ##p \equiv 1 \pmod 4##...