Ken Yamamura

National Defense Academy of Japan | NDAJ · Department of Mathematics

The purpose of this article is to describe simple ways to construct quadratic number fields each having an unramified extension which properly contains the Hilbert class field of its genus field (in the wide sense). The motivation of this study is the author’s observation that under the Generalized Riemann Hypothesis (GRH), for most quadratic numbe...

In the previous paper [15], we determined the structure of the Galois groups Gal(Kur/K) of the maximal unramified extensions Kur of imaginary quadratic number fields K of conductors (Formula Presented) 1000 under the Generalized Riemann Hypothesis (GRH) except for 23 fields (these are of conductors (Formula Presented) 723) and give a table of Gal(K...

We determine the structures of the Galois groups Gal(Kur/K) of the maximal unramified extensions Kur of imaginary quadratic number fields K of conductors ≦ 420 ≦ 719 under the Generalized Riemann Hypothesis). For all such K, Kur is K, the Hilbert class field of K, the second Hilbert class field of K, or the third Hilbert class field of K. The use o...

The maximal unramified extensions of the imaginary quadratic number fields with class number two are determined explicitly under the Generalized Riemann Hypothesis.

In this paper, we determine all the imaginary abelian number fields with class number one. There exist exactly 172 imaginary abelian number fields with class number one. The maximal conductor of these fields is 10921 = 67 · 163, which is the conductor of the biquadratic number field Q(√-67, √-163).

In this paper, we determine all the imaginary abelian number fields with class number one. There exist exactly 172 imaginary abelian number fields with class number one. The maximal conductor of these fields is 10921 = 67 · 163, which is the conductor of the biquadratic number field $\mathbf{Q}(\sqrt{-67}, \sqrt{-163})$.