Structure of $$\mathrm {Gal}(\mathbb {k}_2^{(2)}/\mathbb {k})$$ Gal ( k 2 ( 2 ) / k ) for some fields $$\mathbb {k}=\mathbb {Q}(\sqrt{2p_1p_2},i)$$ k = Q ( 2 p 1 p 2 , i ) with $$\mathbf {C}l_2(\mathbb {k})\simeq (2, 2, 2)$$ C l 2 ( k ) ≃ ( 2 , 2 , 2 )

Abstract

Let $p_1 \equiv p_2 \equiv5\pmod8$ be different primes. Put $i=\sqrt{-1}$ and $d=2p_1p_2$, then the bicyclic biquadratic field $k=Q(\sqrt{d}, \sqrt{-1})$ 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 $\mathrm{Gal}(k_2^{(2)}/k)$ of the second Hilbert 2-class field $k_2^{(2)}$ of $k$. We study the capitulation problem of the 2-classes of $k$ in its seven unramified quadratic extensions $K_i$ and in its seven unramified bicyclic biquadratic extensions $L_i$.