Hilbert genus fields of biquadratic fields

The Hilbert genus field of the real biquadratic field (K = mathbb{Q}left( {sqrt delta ,sqrt d } right)) 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 (K = mathbb{Q}left( {sqrt delta ,sqrt d } right)) , 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.