How Square are the Cube Roots of Unity?

Algebra Level 4

$\Large{z^3=1}$ Let $z_1,z_2,z_3$ be the cube roots of unity satisfying the equation above. If $a\in\{1,2,3\}$ and $b\in\{1,2,3\}$ ,when is it true that $z^{2}_a= z_b$ ?

Only true for

z_a = z_b = 1

True for all

a \neq b

True when

z_a

is real and

a=b

, and true when

z_a

and

z_b

aren't real and

a\neq b

True for all

a

and

b