Does Zp Contain Primitive Fourth Roots of Unity?: Investigating p

[Funky1981]

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?

 

[I like Serena]

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 $p \equiv 1 \pmod 4$...