Let ${\mathrm{z}}^{\mathrm{n}}-1=0$, have n roots given by ${\mathrm{\alpha }}_1,{\mathrm{\alpha }}_2,\dots \dots ,{\mathrm{\alpha }}_{\mathrm{n}}$ then :

$\therefore$For given equation ${\mathrm{z}}^{\mathrm{n}}-1=0$
${\mathrm{\alpha }}_1+{\mathrm{\alpha }}_2+\dots ..+{\mathrm{\alpha }}_{\mathrm{n}}=0\mathrm{\&}{\mathrm{\alpha }}_1{\mathrm{\alpha }}_2\dots \dots {\mathrm{\alpha }}_{\mathrm{n}}=(-1{)}^{\mathrm{n}}$ and each ${\mathrm{\alpha }}_{\mathrm{i}}$ must satisfy the given equation
$\begin{array}{l}{\mathrm{\alpha }}_{\mathrm{i}}^{\mathrm{n}}-1=0\mathrm{\forall }\mathrm{i}=1,2,\dots \dots \mathrm{n}\Rightarrow {\mathrm{\alpha }}_{\mathrm{i}}^{\mathrm{n}}=1\\ \left|{\mathrm{\alpha }}_{\mathrm{i}}\right|=1\mathrm{\forall }\mathrm{i}=1,2,\dots \dots ..\mathrm{n}\end{array}$