Hilbert Genus Fields of Imaginary Biquadratic Fields

Let K_(0)=Q(√δ)beaquadraticfield.Forthose K_(0) withoddclassnumber,much work has been done on the explicit construction of the Hilbert genus field of a biquadratic extension K=Q(√δ,√d)over Q.Whenδ=2 or p with p≡1 mod 4 a prime and K is real,it was described in Yue(Ramanujan J 21:17–25,2010)and Bae and Yue(Ramanujan J 24:161–181,2011).In this paper,we describe the Hilbert genus field of K explicitly when K_(0) is real and K is imaginary.In fact,we give the explicit construction of the Hilbert genus field of any imaginary biquadratic field which contains a real quadratic subfield of odd class number.