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 pure number field generated by a complex root $\alpha$ of a monic irreducible polynomial $F\left(x\right)=x^{2^r·3^k·7^s}-m\in \mathbb{Z}\left[x\right]$, where $r$, $k$, $s$ are three positive natural integers. The purpose of this paper is to study the monogenity of $K$. Our results are illustrated by some examples.

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 $L=K\left(\alpha \right)$ be an extension of a number field $K$, where $\alpha$ satisfies the monic irreducible polynomial $P\left(X\right)=X^p-\beta$ of prime degree belonging to ${𝔬}_K\left[X\right]$ (${𝔬}_K$ is the ring of integers of $K$). The purpose of this paper is to study the monogenity of $L$ over $K$ by a simple and practical version of Dedekind’s criterion characterizing the existence of power integral bases over an arbitrary Dedekind ring by using the Gauss valuation and the index ideal. As an illustration, we determine an integral basis of a pure nonic field $L$ with a...

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 $(2,2,2)$.