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 2000 Mathematics Subject Classification. Primary 11R16, 11R29, 11R32, 11R37; Secondary 20D15.

Coclass of ${\rm Gal}({\mathbb K}_2^{(2)}/{\mathbb K})$ for some fields ${\mathbb K} = {\mathbb Q}(\sqrt{p_1p_2q}, \sqrt{-1})$ with 2-class groups of types (2, 2, 2)

“…Therefore: 4 and G 2 /G 2 is of type (2, 2, 2), since (σρ) 2 = ρ 2 = σ 2 n = (σ 4 ) 2 n−2 (in this case n ≥ 2). To calculate ker V G→G2 , we need the following lemma.…”

“…If ( p2 q ) = −1, then Cl 2 (K 2 ), Cl 2 (K 3 ), Cl 2 (K 6 ) and Cl 2 (K 7 ) are of type (2,4), and they are of type (2, 2, 2) otherwise.…”

“…However, nothing is known about base fields of higher degree, that is why we treat the imaginary bicyclic biquadratic field k = Q( √ d, √ −1). In [7,8,4] we have determined the maximal unramified pro-2 extension of some fields k = Q( √ d, √ −1) with 2-class group of type (2, 2, 2); we also determined the generators and the structure of the non-Abelian Galois group Gal(k (2) 2 /k) of the second Hilbert 2-class field k…”

Unramified extensions of some cyclic quartic fields

“…In [5], the authors studied the capitulation problem of the 2-classes of the biquadratic fields Q( √ p 1 p 2 q, √ −1) with 2-class group isomorphic to Z/2Z×Z/2Z× Z/2Z, in its 14 unramified abelian extension within the first Hilbert 2-class field.…”

The Generators of the $2$-Class Group of Some Fields: Correction to Theorem 3 of }

“…Let k be an algebraic number field and let C k,2 denote its 2-class group, that is the 2-Sylow subgroup of the ideal class group C k of k. The structure and the generators of C k,2 play an important role in Number Theory, in fact they can help to determine the structure and the generators of the maximal unramified pro-2 extension of k, and they also help to solve the capitulation problem of the 2-ideal classes of k in its unramified extensions see [6,7,8,9,10,11,12,16,17]. Let k = Q( √ d, i), where d is a square-free integer.…”