Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 3 / 18H / i / Solution

Choose the radicals with positive imaginary part. The roots of unity areThe first comes from $z^2+z+1=0$. For the second, $t={\zeta }_5+{\zeta }_5^{-1}$ obeys $t^2+t-1=0$, so $t=(\sqrt{5}-1)/2$; solving $z^2-tz+1=0$ gives the displayed expression. The Kummer extension towers areEach step is quadratic and the base contains the second roots of unity. The second tower really has degrees $2,2$: its intermediate field is real, while the added radical is nonreal, and $\sqrt{5}=1+2({\zeta }_5+{\zeta }_5^{-1})$ belongs to the cyclotomic field.