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.

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???