Hilbert Genus Fields of Imaginary Biquadratic Fields

Zhe Zhang , Qin Yue

Communications in Mathematics and Statistics ›› 2017, Vol. 5 ›› Issue (2) : 175 -197.

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Communications in Mathematics and Statistics ›› 2017, Vol. 5 ›› Issue (2) : 175 -197. DOI: 10.1007/s40304-017-0107-8
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Hilbert Genus Fields of Imaginary Biquadratic Fields

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Abstract

Let $K_0={\mathbb {Q}}\left( \sqrt{\delta }\right) $ be a quadratic field. For those $K_0$ with odd class number, much work has been done on the explicit construction of the Hilbert genus field of a biquadratic extension $K={\mathbb {Q}}\left( \sqrt{\delta },\sqrt{d}\right) $ over ${\mathbb {Q}}$. When $\delta =2$ or p with $p\equiv 1\bmod 4$ 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.

Keywords

Class group / Hilbert symbol / Hilbert genus fields

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Zhe Zhang, Qin Yue. Hilbert Genus Fields of Imaginary Biquadratic Fields. Communications in Mathematics and Statistics, 2017, 5(2): 175-197 DOI:10.1007/s40304-017-0107-8

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Funding

National Natural Science Foundation of China(11501429)

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