Narrow Results

Let p₁ ≡ p₂ ≡ -q ≡ 1 (mod 4) be primes such that and . Put and d = p₁p₂q, then the bicyclic biquadratic field 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 of the second Hilbert 2-class field of 𝕂, we study the 2-class field tower of 𝕂, and we study the capitulation problem of the 2-classes of 𝕂 in its fourteen abelian unramified extensions of relative degrees two and four.

Let p₁ ≡ p₂≡ -q ≡ 1 (mod 4) be different primes such that . Put d = p₁p₂q and , then the bicyclic biquadratic field 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 14 unramified abelian extensions 𝕂_j and 𝕃_j within , 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 of the second Hilbert 2-class field of 𝕂. Additionally, the abelian type invariants of the groups Cl₂(𝕂_j) and Cl₂(𝕃_j) and the length of the 2-class tower of 𝕜 are given.

We construct an infinite family of imaginary bicyclic biquadratic number fields 𝕜 with the 2-ranks of their 2-class groups are ≥ 3, whose strongly ambiguous classes of 𝕜/ℚ(i) capitulate in the absolute genus field 𝕜^(*), which is strictly included in the relative genus field (𝕜/ℚ(i))* and we study the capitulation of the 2-ideal classes of 𝕜 in its quadratic extensions included in 𝕜^(*).

The capitulation problem is one of the most important topic in number theory, and as it is closely related to the group theory, we present, in this paper, some group theoretical results to solve this problem, in a particular case, whenever $G/G^{\prime }\simeq (2,2,2)$ for some metabelian $2$-group $G$. Then we illustrate our results by some examples.

Let $k$ be a number field and $k_2^{(1)}$ (respectively, $k_2^{(2)}$) its first (respectively, second) Hilbert $2$-class field. Let $G=\mathrm{\text{Gal}}(k_2^{(2)}/k)$ be the Galois group of $k_2^{(2)}/k$. The purpose of this note is to determine the structure of $G$ for some special Dirichlet fields $k=\mathbb{Q}(\sqrt{d},\sqrt{-1})$.