Principalization of 2-class groups of type (2, 2, 2) of biquadratic fields ${\mathbb Q}(\sqrt{p_1p_2q}, \sqrt{-1})$

Abstract

Let p1 ≡ p2≡ -q ≡ 1 (mod 4) be different primes such that [Formula: see text]. Put d = p1p2q and [Formula: see text], then the bicyclic biquadratic field [Formula: see text] has an elementary abelian 2-class group, Cl2(𝕜), of rank 3. In this paper, we study the principalization of the 2-classes of 𝕜 in its 14 unramified abelian extensions 𝕂j and 𝕃j within [Formula: see text], that is the Hilbert 2-class field of 𝕜. We determine the nilpotency class, the coclass, generators and the structure of the metabelian Galois group [Formula: see text] of the second Hilbert 2-class field [Formula: see text] of 𝕂. Additionally, the abelian type invariants of the groups Cl2(𝕂j) and Cl2(𝕃j) and the length of the 2-class tower of 𝕜 are given.