Coclass of ${\rm Gal}({\mathbb K}_2^{(2)}/{\mathbb K})$ for some fields ${\mathbb K} = {\mathbb Q}(\sqrt{p_1p_2q}, \sqrt{-1})$ with 2-class groups of types (2, 2, 2)

Abstract

Let p1 ≡ p2 ≡ -q ≡ 1 ( mod 4) be primes such that [Formula: see text] and [Formula: see text]. Put [Formula: see text] and d = p1p2q, then the bicyclic biquadratic field [Formula: see text] has an elementary Abelian 2-class group of rank 3. In this paper we determine the nilpotency class, the coclass, the generators and the structure of the non-Abelian Galois group [Formula: see text] of the second Hilbert 2-class field [Formula: see text] of 𝕂, we study the 2-class field tower of 𝕂, and we study the capitulation problem of the 2-classes of 𝕂 in its fourteen abelian unramified extensions of relative degrees two and four.