Abstract
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.
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References
Bae S, Yue Q. Hilbert genus fields of real biquadratic fields. Ramanujan J, 2011, 24: 161–181
Conner P E, Hurrelbrink J. Class Number Parity, Ser Pure Math 8. Singapore: World Scientific, 1988
Herglotz G. Über einen Dirichletschen Satz. Math Z, 1922, 12: 225–261
Lang S. Cyclotomic Fields I and II. GTM 121. New York: Springer-Verlag, 1990
Neukirch J. Class Field Theory. Berlin-Heidelberg-New York-Tokyo: Springer-Verlag, 1986
Sime P. Hilbert class fields of real biquadratic fields. J Number Theory, 1995, 50: 154–166
Yue Q. The generalized Rédei matrix. Math Z, 2009, 261: 23–37
Yue Q. Genus fields of real biquadratic fields. Ramanujan J, 2010, 21: 17–25
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Ouyang, Y., Zhang, Z. Hilbert genus fields of biquadratic fields. Sci. China Math. 57, 2111–2122 (2014). https://doi.org/10.1007/s11425-014-4867-2
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DOI: https://doi.org/10.1007/s11425-014-4867-2