= Solution
Choose the radicals with positive imaginary part. The <roots of unity> are
$$
\boxed{\zeta_3=\frac{-1+\sqrt{-3}}2,\qquad
\zeta_5=\frac{\sqrt5-1+\sqrt{-10-2\sqrt5}}4.}
$$
The first comes from $z^2+z+1=0$. For the second, $t=\zeta_5+\zeta_5^{-1}$ obeys $t^2+t-1=0$, so $t=(\sqrt5-1)/2$; solving $z^2-tz+1=0$ gives the displayed expression. The <Kummer extension> towers are
$$
\mathbb Q\subset\mathbb Q(\sqrt{-3})=\mathbb Q(\mu_3),
\qquad
\mathbb Q\subset\mathbb Q(\sqrt5)\subset
\mathbb Q(\sqrt5,\sqrt{-10-2\sqrt5})=\mathbb Q(\mu_5).
$$
Each 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 $\sqrt5=1+2(\zeta_5+\zeta_5^{-1})$ belongs to the cyclotomic field.
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