Hilbert genus fields of real biquadratic fields

The Hilbert genus field of the real biquadratic field $$K=\mathbb {Q}(\sqrt{\delta },\sqrt{d})$$ is described by Yue (Ramanujan J 21:17–25, 2010) and by Bae and Yue (Ramanujan J 24:161–181, 2011) explicitly in the case $$\delta =2$$ or $$p$$ with $$p\equiv 1 \, \mathrm{mod}\, 4$$ a prime and $$d$$ a squarefree positive integer. In this article, we describe explicitly the case that $$\delta =p, 2p$$ or $$p_1p_2$$ where $$p$$ , $$p_1$$ , and $$p_2$$ are primes congruent to $$3$$ modulo $$4$$ , and $$d$$ is any squarefree positive integer, thus complete the construction of the Hilbert genus field of real biquadratic field $$K=K_0(\sqrt{d})$$ such that $$K_0=\mathbb {Q}(\sqrt{\delta })$$ has an odd class number.