Assuming the Generalized Riemann Hypothesis (GRH), we show that the norm-Euclidean Galois cubic fields are exactly those with discriminant$$\Delta =7^2,9^2,{13}^2,{19}^2,{31}^2,{37}^2,{43}^2,{61}^2,{67}^2,{103}^2,{109}^2,{127}^2,{157}^2.$$A large part of the proof is in establishing the following more general result: Let $K$ be a Galois number field of odd prime degree $\ell$ and conductor $f$. Assume the GRH for ${\zeta }_K\left(s\right)$. If$$38{(\ell -1)}^2{(logf)}^{6}loglogf\<f,$$then $K$ is not norm-Euclidean.

On démontre, à partir de résultats de H.J. Godwin, H. Brunotte et F. Halter-Koch, le théorème suivant : soit $K$ un corps cubique cyclique de conducteur $m$ dont le groupe de Galois $G$ est engendré par $\sigma$; soit $E$ le groupe des unités de norme 1.Soit $ϵ\in E$, $ϵ\ne 1$, telle que $𝒮\left(ϵ\right)=\frac{1}{2}\left[\right(ϵ-{ϵ}^{\sigma }{)}^2+({ϵ}^{\sigma }-{ϵ}^{{\sigma }^2}{)}^2+({ϵ}^{{\sigma }^2}-ϵ{)}^2]$ soit minimum. Alors $ϵ$ est un $\mathbf{Z}\left[G\right]$-générateur de $E$.

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).

In this paper we introduce multiplicative lattices in ${\left({ℝ}^{\>0}\right)}^r$ and determine finite unions of suitable simplices as fundamental domains for sublattices of finite index. For this we define cyclic non-negative bases in arbitrary lattices. These bases are then used to calculate Shintani cones in totally real algebraic number fields. We mainly concentrate our considerations to lattices in two and three dimensions corresponding to cubic and quartic fields.

Let $K=\mathbb{Q}\left(\alpha \right)$ be a number field generated by a complex root $\alpha$ of a monic irreducible polynomial $f\left(x\right)=x^{18}-m$, $m\ne \mp 1$, is a square free rational integer. We prove that if $m\equiv 2$ or $3(\mathrm{mod}4)$ and $m\neg \equiv \mp 1(\mathrm{mod}9)$, then the number field $K$ is monogenic. If $m\equiv 1(\mathrm{mod}4)$ or $m\equiv 1(\mathrm{mod}9)$, then the number field $K$ is not monogenic.

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 $(2,2,2)$.

We study a certain finitely generated multiplicative subgroup of the Hilbert class field of a quartic CM field. It consists of special values of certain theta functions of genus 2 and is analogous to the group of Siegel units. Questions of integrality of these specials values are related to the arithmetic of the Siegel moduli space.