Narrow Results

Let $n\ge 3$ be an integer and $d$ an odd square-free integer. We compute the rank of the 2-class group of some fields of the form $L_{n,d}=\mathbb{Q}({\zeta }_{2^n},\sqrt{d})$ when all the prime divisors of $d$ are congruent to $\pm 3(\mathrm{mod}8)$ or $9(\mathrm{mod}16)$.

In this paper, we investigate the cyclicity of the $2$-class group of the first Hilbert $2$-class field of some quadratic number field whose discriminant is not a sum of two squares. For this, let $p_1\equiv p_2\equiv -q\equiv 1(mod4)$ be different prime integers. Put $𝕜=\mathbb{Q}(\sqrt{p_1{p}_{2}q})$, and denote by ${\mathrm{\text{C}}}_{𝕜,2}$ its $2$-class group and by ${𝕜}_2^{(1)}$ (respectively ${𝕜}_2^{(2)}$) its first (respectively second) Hilbert $2$-class field. Then, we are interested in studying the metacyclicity of $G=\mathrm{\text{Gal}}({𝕜}_2^{(2)}/𝕜)$ and the cyclicity of $\mathrm{\text{Gal}}({𝕜}_2^{(2)}/{𝕜}_2^{(1)})$ whenever the $4$-rank of ${\mathrm{\text{C}}}_{𝕜,2}$ is $1$.

Let $d$ be an odd positive square-free integer. In this paper, we shall investigate the structure of the $2$-class group of the cyclotomic ${\mathbb{Z}}_2$-extension of the imaginary biquadratic number field $\mathbb{Q}(\sqrt{d},\sqrt{-1})$ if $d$ is of specifiic form. Furthermore, we deduce the structure of the $2$-class group of the cyclotomic ${\mathbb{Z}}_2$-extension of $\mathbb{Q}(\sqrt{-d})$.

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^(*).

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 $k=\mathbb{Q}(\sqrt{d},\sqrt{-1})$.