Structure of $Gal(k_2^{(2)}/k)$ for some fields $k=Q(\sqrt{2p_1{p}_2},\sqrt{-1})$

Abstract

Let $p_1\equiv p_2\equiv 5(\mathrm{mod}8)$ be different primes. Put $i=\sqrt{-1}$ and $d=2p_1{p}_2$, 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 $L_i$.

Cited by in corpus (1)

• On some metabelian 2-group whose abelianization is of type (2, 2, 2) and applications