Hilbert Genus Fields of Imaginary Biquadratic Fields

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

Funding

National Natural Science Foundation of China(11501429)