On some imaginary triquadratic number fields $k$ with $\mathrm{Cl}_2(k)\simeq(2, 4)$ or $(2, 2, 2)$

On the rank of the $2$-class group of some imaginary triquadratic number fields

Let $d$ be an odd square-free integer and $ζ_8$ a primitive $8$-th root of unity. The purpose of this paper is to investigate the rank of the $2$-class group of the fields $L_d=\mathbb{Q}(ζ_8,\sqrt{d})$.

Triquadratic p-Rational Fields

In his work about Galois representations, Greenberg conjectured the existence, for any odd prime p and any positive integer t, of a multiquadratic p-rational number field of degree 2 t. In this article, we prove that there exists infinitely many primes p such that the triquadratic field Q(p(p + 2), p(p -- 2), i) is p-rational. To do this, we use an analytic result, proved apart in section \S4, providing us with infinitely many prime numbers p such that p + 2 et p -- 2 have ''big'' square factors. Therefore the related imaginary quadratic subfields Q(i $\sqrt$ p + 2), Q(i $\sqrt$ p -- 2) and Q(…

通过双二次域和三二次域构造格

Constructions of lattices via biquadratic and triquadratic fields

本研究利用代数数论工具，证明$D_4$-格可由无穷全虚双二次域族构造，并扩展方法证明$D_8$与$E_8$格可通过该族衍生的三二次域实现，为格构造提供了新的代数数论途径。

The 2-Iwasawa module over certain octic elementary fields

The aim of this paper is to determine the structure of $2$-Iwasawa module of some imaginary triquadratic fields.

Growth of Odd Torsion Over Imaginary Quadratic Fields of Class Number 1

Let $K$ be a non-cylotomic imaginary quadratic field of class number 1 and $E/K$ is an elliptic curve with $E(K)[2]\simeq \mathbb{Z}_1.$ We determine the odd-order torsion groups that can arise as $E(L)_{\text{tor}}$ where $L$ is a quadratic extension of $K.$

On the $p$-rationality of consecutive quadratic fields

In 2016, in the work related to Galois representations, Greenberg conjectured the existence of multi-quadratic $p$-rational number fields of degree $2^{t}$ for any odd prime number $p$ and any integer $t \geq 1$. Using the criteria provided by him to check $p$-rationality for abelian number fields, certain infinite families of quadratic, biquadratic and triquadratic $p$-rational fields have been shown to exist in recent years. In this article, for any integer $k \geq 1$, we build upon the existing work and prove the existence of infinitely many prime numbers $p$ for which the imaginary quadrat…