  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 $m\geq 3$ is a positive
integer and $d$ is a square-free integer whose prime divisors are congruent to
$\pm 3\pmod 8$ or $9\pmod{16}$.
