The equation
\(x^{3}+x-1=0\)
has roots \(\alpha, \beta, \gamma\). Use the relation \(x=\sqrt{ } y\) to show that the equation
\(y^{3}+2 y^{2}+y-1=0\)
has roots \(\alpha^{2}, \beta^{2}, \gamma^{2}\).

Let \(S_{n}=\alpha^{n}+\beta^{n}+\gamma^{n}\).
(i) Write down the value of \(S_{2}\) and show that \(S_{4}=2\).

(ii) Find the values of \(S_{6}\) and \(S_{8}\).