English

Abstract

Let $$\mathbb{k} = \mathbb{Q} \left( {\sqrt 2 ,\; \sqrt d } \right)$$ be an imaginary bicyclic biquadratic number field, where d is an odd negative square-free integer and $$\mathbb{k}_2^{\left( 2 \right)}$$ its second Hilbert 2-class field. Denote by $$G = {\rm{Gal}}\left( {\mathbb{k}_2^{\left( 2 \right)}/ \mathbb{k}} \right)$$ the Galois group of $${\mathbb{k}_2^{\left( 2 \right)}/ \mathbb{k}}$$ . The purpose of this note is to investigate the Hilbert 2-class field tower of $$\mathbb{k}$$ and then deduce the structure of G.