Related papers: Normes cyclotomiques na{\"i}ves et unit{\'e}s loga…

We use $\ell$-adic class field theory to take a new view on cyclotomic norms and Leopoldt or Gross generalized conjectures. By the way we recall and complete some classical results. We illustrate the logarithmic approach by various…

We examine when units in a field are the norms of elements in an extension field, given certain conditions. We apply these results to the study of the $\ell$-class groups in lifts of the anti-cyclotomic $\mathbb{Z}_2$-extension of…

Let $\ell$ be a rational prime number. Assuming the Gross-Kuz'min conjecture along a $\Zl$-extension $K\_{\infty}$ of a number field $K$, we show that there exist integers $\mut$, $\lat$ and $\widetilde{\nu}$ such that the exponent…

For a finite group $G$, we introduce a generalization of norm relations in the group algebra $\mathbb Q[G]$. We give necessary and sufficient criteria for the existence of such relations and apply them to obtain relations between the…

Let $\mathbf{Was}(\mathbb{K})$ denote the group of Washington's cyclotomic units of any abelian number field $\mathbb{K}$. If $\mathbb{K}$ coincides with its genus field in the narrow sense, we give a $\Lambda$-basis of…

Let $A' = \varprojlim_n A'_n$ be the projective limit of the $p$-parts of the ideal class groups of the $p$ integers in the $\mathbb{Z}_p$-cyclotomic extension $K_{\infty}/K$ of a CM number field $K$. We prove in this paper that the…

For an algebraic number field K such that prime l splits completely in K we define a regulator R(K) that characterize the subgroup of universal norms from the cyclotomic extension of K in the completed group of S-units of K, where S…

We investigate the norm maps of algebraic even $K$-groups of finite extensions of number fields. Namely, we show that they are surjective in most situations. In the event that they are not surjective, we give a criterion in determining when…

Given a number field $K$, a finite abelian group $G$ and finitely many elements $\alpha_1,\ldots,\alpha_t\in K$, we construct abelian extensions $L/K$ with Galois group $G$ that realise all of the elements $\alpha_1,\ldots,\alpha_t$ as…

We investigate the group of universal norms attached to the cyclotomic Z {\ell}-tower of a totally real number field in connection with Grenberg's conjecture on Iwasawa invariants of such a field.

For a number field $K$ let $\mathcal{S}_K$ be the maximal subgroup of the multiplicative group $K^\times$ that embeds into the unit circle under each embedding of $K$ into the complex numbers. The group $\mathcal{S}_K$ can be seen as an…

Let $\ell$ be a rational prime number and $K$ a number field. We prove that the logarithmic module $X_{d}$ attached to a $\mathbb{Z}_{\ell}^{d}$-extension $K_{d}$ of $K$ is a noetherian $\Lambda_{d}$-module. Moreover, under the…

For a real abelian field and for an odd prime p splitting in the field, we study a map between the p-parts of the class group and of the quotient of units modulo Cyclotomic Units, respectively, along the cyclotomic Z_p-extension of the…

In this article we prove that (1-zeta+zeta^2) is a unit in the ring of integers of the cyclotomic field where zeta is a primitive n-th root of unity and n is coprime to 2 and 3. We also prove that for prime n,…

Let $A' = \varprojlim_n$ be the projective limit of the $p$-parts of the ideal class groups of the $p$ integers in the $\Z_p$-cyclotomic extension $\K_{\infty}/\K$ of a CM number field $\K$. We prove in this paper that the $T$ part…

We call a (q-1)-th Kummer extension of a cyclotomic function field a quasi-cyclotomic function field if it is Galois, but non-abelian, over the rational function field with the constant field of q elements. In this paper, we determine the…

Upon quotienting by units, the elements of norm 1 in a number field $K$ form a countable subset of a torus of dimension $r_1 + r_2 - 1$ where $r_1$ and $r_2$ are the numbers of real and pairs of complex embeddings. When $K$ is Galois with…

This paper extends the study of rank-metric codes in extension fields $\mathbb{L}$ equipped with an arbitrary Galois group $G = \mathrm{Gal}(\mathbb{L}/\mathbb{K})$. We propose a framework for studying these codes as subspaces of the group…

We give an analytic proof of the norm index theorem $[I_:K^* N(I_L)] =[L:K]$ for cyclic extensions of number fields using spectral theory of the idele class group.

Let $H= \mathbb{Q}(\zeta_{n} + {\zeta_{n}}^{-1})$ and $\ell$ be an odd prime such that $q \equiv 1 \pmod \ell$ for some prime factor $q$ of $n$. We get a bound on the $\ell$-rank of the class group of $H$(under some conditions) in terms of…