Hilbert genus fields of biquadratic fields

The Hilbert genus field of the real biquadratic field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K = \mathbb{Q}\left( {\sqrt \delta ,\sqrt d } \right)$$\end{document} is described by Yue (2010) and Bae and Yue (2011) explicitly in the case δ = 2 or p with p ≡ 1 mod 4 a prime and d a squarefree positive integer. In this article, we describe explicitly the Hilbert genus field of the imaginary biquadratic field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K = \mathbb{Q}\left( {\sqrt \delta ,\sqrt d } \right)$$\end{document}, where δ = −1,−2 or −p with p ≡ 3mod 4 a prime and d any squarefree integer. This completes the explicit construction of the Hilbert genus field of any biquadratic field which contains an imaginary quadratic subfield of odd class number.