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Search: All articles in the CMB digital archive with keyword units

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1. CMB 2011 (vol 56 pp. 148)

Oukhaba, Hassan; Viguié, Stéphane
On the Gras Conjecture for Imaginary Quadratic Fields
In this paper we extend K. Rubin's methods to prove the Gras conjecture for abelian extensions of a given imaginary quadratic field $k$ and prime numbers $p$ that divide the number of roots of unity in $k$.

Keywords:elliptic units, Stark units, Gras conjecture, Euler systems
Categories:11R27, 11R29, 11G16

2. CMB 2011 (vol 54 pp. 330)

Mouhib, A.
Sur la borne inférieure du rang du 2-groupe de classes de certains corps multiquadratiques
Soient $p_1,p_2,p_3$ et $q$ des nombres premiers distincts tels que $p_1\equiv p_2\equiv p_3\equiv -q\equiv 1 \pmod{4}$, $k = \mathbf{Q} (\sqrt{p_1}, \sqrt{p_2}, \sqrt{p_3}, \sqrt{q})$ et $\operatorname{Cl}_2(k)$ le $2$-groupe de classes de $k$. A. Fröhlich a démontré que $\operatorname{Cl}_2(k)$ n'est jamais trivial. Dans cet article, nous donnons une extension de ce résultat, en démontrant que le rang de $\operatorname{Cl}_2(k)$ est toujours supérieur ou égal à $2$. Nous démontrons aussi, que la valeur $2$ est optimale pour une famille infinie de corps $k$.

Keywords:class group, units, multiquadratic number fields
Categories:11R29, 11R11

3. CMB 2002 (vol 45 pp. 428)

Mollin, R. A.
Criteria for Simultaneous Solutions of $X^2 - DY^2 = c$ and $x^2 - Dy^2 = -c$
The purpose of this article is to provide criteria for the simultaneous solvability of the Diophantine equations $X^2 - DY^2 = c$ and $x^2 - Dy^2 = -c$ when $c \in \mathbb{Z}$, and $D \in \mathbb{N}$ is not a perfect square. This continues work in \cite{me}--\cite{alfnme}.

Keywords:continued fractions, Diophantine equations, fundamental units, simultaneous solutions
Categories:11A55, 11R11, 11D09

4. CMB 2000 (vol 43 pp. 60)

Farkas, Daniel R.; Linnell, Peter A.
Trivial Units in Group Rings
Let $G$ be an arbitrary group and let $U$ be a subgroup of the normalized units in $\mathbb{Z}G$. We show that if $U$ contains $G$ as a subgroup of finite index, then $U = G$. This result can be used to give an alternative proof of a recent result of Marciniak and Sehgal on units in the integral group ring of a crystallographic group.

Keywords:units, trace, finite conjugate subgroup
Categories:16S34, 16U60

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