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Let $p\equiv 1(mod8)$ and $q\equiv 3(mod8)$ be two prime integers and let $\ell \notin \{-1,p,q\}$ be a positive odd square-free integer. Assuming that the fundamental unit of $\mathbb{Q}\left(\sqrt{2p}\right)$ has a negative norm, we investigate the unit group of the fields $\mathbb{Q}(\sqrt{2},\sqrt{p},\sqrt{q},\sqrt{-\ell })$.

Let $d$ be a square free integer and $L_d:=\mathbb{Q}({\zeta }_8,\sqrt{d})$. In the present work we determine all the fields $L_d$ such that the $2$-class group, ${\mathrm{Cl}}_2\left(L_d\right)$, of $L_d$ is of type $(2,4)$ or $(2,2,2)$.

Let $𝕜=\mathbb{Q}\left(\sqrt{2},\sqrt{d}\right)$ be an imaginary bicyclic biquadratic number field, where $d$ is an odd negative square-free integer and ${𝕜}_2^{\left(2\right)}$ its second Hilbert $2$-class field. Denote by $G=\mathrm{Gal}({𝕜}_2^{\left(2\right)}/𝕜)$ the Galois group of ${𝕜}_2^{\left(2\right)}/𝕜$. The purpose of this note is to investigate the Hilbert $2$-class field tower of $𝕜$ and then deduce the structure of $G$.

Let $D$ be an integral domain with the quotient field $K$, $X$ an indeterminate over $K$ and $x$ an element of $D$. The Bhargava ring over $D$ at $x$ is defined to be ${𝔹}_x\left(D\right):=\{f\in K\left[X\right]:\text{for}\text{all}a\in D,f(xX+a)\in D\left[X\right]\}$. In fact, ${𝔹}_x\left(D\right)$ is a subring of the ring of integer-valued polynomials over $D$. In this paper, we aim to investigate the behavior of ${𝔹}_x\left(D\right)$ under localization. In particular, we prove that ${𝔹}_x\left(D\right)$ behaves well under localization at prime ideals of $D$, when $D$ is a locally finite intersection of localizations. We also attempt a classification of integral domains $D$...

Let $k$ be a number field with a 2-class group isomorphic to the Klein four-group. The aim of this paper is to give a characterization of capitulation types using group properties. Furthermore, as applications, we determine the structure of the second 2-class groups of some special Dirichlet fields $𝕜=\mathbb{Q}(\sqrt{d},\sqrt{-1})$, which leads to a correction of some parts in the main results of A. Azizi and A. Zekhini (2020).

Let $d$ be an odd square-free integer, $m\ge 3$ any integer and $L_{m,d}:=\mathbb{Q}({\zeta }_{2^m},\sqrt{d})$. In this paper, we shall determine all the fields $L_{m,d}$ having an odd class number. Furthermore, using the cyclotomic ${\mathbb{Z}}_2$-extensions of some number fields, we compute the rank of the $2$-class group of $L_{m,d}$ whenever the prime divisors of $d$ are congruent to $3$ or $5(mod8)$.