Question

Aratati ca singurele numere complexe Z cu proprietatea ca z^{4} =1 sunt 1,-1,i si -i

Answer 1

$z^4 = 1 \\ \\ Notam z^2 = t. \\ \\ t^2 = 1 \Rightarrow t = \pm 1 \\ \\ \boxed{1}\quad t=1 \Rightarrow z^2 = 1 \Rightarrow z = \pm1 \Rightarrow z\in \Big\{-1,1\Big\} \\ \\ \boxed{2}\quad t = -1 \Rightarrow z^2 = -1 \Rightarrow z = \pm i \Rightarrow z\in \Big\{-i,i\Big\} \\ \\ Din \boxed{1} \cup \boxed{2} \Rightarrow \boxed{S= \Big\{-1,1,-i,i\Big\}}$