Hilbert genus fields of some number fields with high degrees

The aim of this paper is to give some properties of Hilbert genus fields and construct the Hilbert genus fields of the fields $L_{m,d}:=\mathbb{Q}({\zeta }_{2^m},\sqrt{d})$, where $m\ge 3$ is a positive integer and $d$ is a square-free integer whose prime divisors are congruent to $\pm 3(\mathrm{m}\mathrm{o}\mathrm{d}8)$ or $9(\mathrm{m}\mathrm{o}\mathrm{d}16)$.