  Let $p$ be a prime number such that $p=2$ or $p\equiv 1\pmod 4$. Let
$\varepsilon_p$ 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 $\mathbb{Q}(\sqrt{-a\varepsilon_p\sqrt{p}})$.
