Principalization of 2-class groups of type (2,2,2) of biquadratic fields ℚ(√(\strut p₁p₂q),√(\strut -1))

2014/04/14 by Abdelmalek Azizi, Abdelkader Zekhnini, Azizi, Abdelmalek +5
Mathematics · #11R29 #11R32 #11R37 #FOS: Mathematics #Number Theory (math.NT) #Primary 11R16 #Secondary 20D15 #math.NT #msc:11R16 #msc:11R29 #msc:11R32 #msc:11R37 #msc:20D15

paper · pdf · doi:10.48550/arxiv.1404.3761

arxiv created 2014/04/14 · arxiv updated 2014/04/16

Abstract

Let p₁≡ p₂≡ -q≡1 \pmod4 be different primes such that ((2)/(p₁))= ((2)/(p₂))=((p₁)/(q))=((p₂)/(q))=-1. Put d=p₁p₂q and i=√(-1), then the bicyclic biquadratic field k=ℚ(√(d),i) 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 fourteen unramified abelian extensions \mathbbK_j and \mathbbL_j within k₂⁽¹⁾, 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 G=Gal(\mathbbL/k) of the second Hilbert 2-class field k₂⁽²⁾ of k. Additionally, the abelian type invariants of the groups Cl₂(\mathbbK_j) and Cl₂(\mathbbL_j) and the length of the 2-class tower of k are given.

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