Abdelmalek Azizi, Abdelkader Zekhnini, Mohammed Taous

COCLASS OF Gal(k^2_2/k) FOR SOME FIELDS k=Q(\sqrt{ p_1p_2q}, \sqrt{ -1}) WITH 2-CLASS GROUPS OF TYPES (2, 2, 2)
Let p_1 \equiv p_2 \equiv-q\equiv1\pmod4 be primes such that $\left(\dfrac{2}{p_1}\right)=1$ and $\left(\dfrac{2}{p_2}\right)=\left(\dfrac{p_1}{p_2}\right)=\left(\dfrac{p_1}{q}\right)=-1$. Put $i=\sqrt{-1}$ and $d=p_1p_2q$, then the bicyclic biquadratic field $\kk=\K$ 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 $\mathrm{Gal}(\L/\kk)$ of the second Hilbert 2-class field $\L$ of $\kk$, we study the 2-class field tower of $\kk$, and we study the capitulation problem of the 2-classes of $\kk$ in its fourteen abelian unramified extensions of relative degrees two and four.
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