  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. In the present paper, we construct the
Hilbert genus field of the real cyclic quartic fields
$\mathbb{Q}(\sqrt{a\varepsilon_p\sqrt{p}})$.
