Hilbert genus fields of real biquadratic fields

2015 
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.
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