Question

If the fourth roots of units are $z_1,z_2,z_3,z_4$, then $z_1^2+z_2^2+z_3^2+z_4^2$ is equal to

• A. 1
• B. 0
• C. i
• D. none of these

Answer 1

The correct option is B 0

Hint: $z^4=1,z=(cos2\pi +isin2r\pi {)}^{\frac{1}{4}}$
$r=0,1,2,3$
$z=1,e^{\frac{2\pi }{4}},e^{\frac{4\pi i}{4}}{e}^{\frac{6\pi i}{4}}$
solution:$z^4=1$
$z=1,e^{\frac{2\pi }{2}},e^{\pi i},e^{\frac{3\pi i}{2}}$
$z=1$
$z_1=1$
$z_2=i$
$z_3=-1$
$z_4=-i$
$z_1^2+z_2^2+z_3^2+z_4^2=1^2+i^2+1+(-i{)}^2=0$