In this paper we consider a subfield $K$ in a cyclotomic field $k_m$ of conductor $m$ such that $\left[k_m : K\right] = 2$ in the cases of $m = \ell p^n$ with a prime $p,$ where $\ell = 4$ or $p > \ell = 3.$ Then the theme is to know whether the ring of integers in $K$ has a power basis or does not.
"Monogenesis of the rings of integers in certain imaginary abelian fields." Nagoya Math. J. 168 85 - 92, 2002.