(a) Show that (x^{5}-1=(x-1)left(x^{4}+x^{3}+x^{2}+x+1 ight)). Deduce that if (omega=e^{2 pi i / 5}) then (omega^{4}+omega^{3}+omega^{2}+omega+1=0). (b) Let...

(a) Show that \(x^{5}-1=(x-1)\left(x^{4}+x^{3}+x^{2}+x+1\right)\). Deduce that if \(\omega=e^{2 \pi i / 5}\) then \(\omega^{4}+\omega^{3}+\omega^{2}+\omega+1=0\).

(b) Let \(\alpha=2 \cos \frac{2 \pi}{5}\) and \(\beta=2 \cos \frac{4 \pi}{5}\). Show that \(\alpha=\omega+\omega^{4}\) and \(\beta=\omega^{2}+\omega^{3}\). Find a quadratic equation with roots \(\alpha, \beta\). Hence show that

\[ \cos \frac{2 \pi}{5}=\frac{1}{4}(\sqrt{5}-1) \]