On Hilbert genus fields of imaginary cyclic quartic fields

Let $p$ be a prime number such that $p=2$ or $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$. Let ${\epsilon }_p$ denote the fundamental unit of $\mathbb{Q}(\sqrt{p})$ and let $a$ be a positive square-free integer. The main aim of this paper is to determine explicitly the Hilbert genus field of the imaginary cyclic quartic fields of the form $\mathbb{Q}(\sqrt{-a{\epsilon }_p\sqrt{p}})$.