If $K/k$ is a finite Galois extension of number fields with Galois group $G$, then the kernel of the capitulation map $C{l}_k\to C{l}_K$ of ideal class groups is isomorphic to the kernel $X\left(H\right)$ of the transfer map $H/H^{\text{'}}\to A,$ where $H=\text{Gal}(\tilde{K}/k),A=\text{Gal}(\tilde{K}/K)$ and $\tilde{K}$ is the Hilbert class field of $K$. H. Suzuki proved that when $G$ is abelian, $\left|G\right|$ divides $\left|X\right(H\left)\right|$. We call a finite abelian group $X$ a transfer kernel for $G$ if $X\cong X\left(H\right)$ for some group extension $A\hookrightarrow H\twoheadrightarrow G$. After characterizing transfer kernels in terms of integral representations of $G$, we show that $X$ is a transfer kernel for...

In this paper we study the groups of circular numbers and circular units in Sinnott’s sense in real abelian fields with exactly four ramified primes under certain conditions. More specifically, we construct $\mathbb{Z}$-bases for them in five special infinite families of cases. We also derive some results about the corresponding module of relations (in one family of cases, we show that the module of Ennola relations is cyclic). The paper is based upon the thesis [6], which builds upon the results of the paper...

This first part of this paper gives a proof of the main conjecture of Iwasawa theory for abelian base fields, including the case $p=2$, by Kolyvagin’s method of Euler systems. On the way, one obtains a general result on local units modulo circular units. This is then used to deduce theorems on the order of $\chi$-parts of $p$-class groups of abelian number fields: first for relative class groups of real fields (again including the case $p=2$). As a consequence, a generalization of the Gras conjecture is stated...

Using both class field and Kummer theories, we propose calculations of orders of two Selmer groups, and compare them: the quotient of the orders only depends on local criteria.

Le but de cet article est d’expliquer comment calculer exactement le nombre de classes d’isomorphismes d’extensions abéliennes de $\mathbb{Q}$ en degré inférieur ou égal à $4$ et de discriminant majoré par une borne donnée. On parvient par exemple à calculer le nombre de corps cubiques cycliques de discriminant inférieur ou égal à ${10}^{37}$.

Let $M$ be a given real quadratic field. We give a fast algorithm for determining all dihedral quartic fields $K$ with mixed signature having power integral bases and containing $M$ as a subfield. We also determine all generators of power integral bases in $K$. Our algorithm combines a recent result of Kable [9] with the algorithm of Gaál, Pethö and Pohst [6], [7]. To illustrate the method we performed computations for $M=\mathbb{Q}\left(\sqrt{2}\right),\mathbb{Q}\left(\sqrt{3}\right),\mathbb{Q}\left(\sqrt{5}\right).$

Soit $k$ une extension algébrique du corps des nombres rationnels, galoisienne et de degré premier $\ell$. Si ${\theta }_0,{\theta }_1,...,{\theta }_{\ell -1}$ désignent des éléments primitifs conjugués de $k$, on note $\overline{{\theta }_{u,j}}$, $j=1,2,...,\ell -1$, leurs résolvantes de Lagrange. Les nombres ${\mu }_j={\overline{{\theta }_{u,j}}}^{\ell }$ sont des éléments primitifs conjugués du corps $C\left(\ell \right)$ des racines $\ell$-ièmes de l’unité.La première partie est consacrée à la caractérisation de ces $\mu$, on en déduit une paramétrisation des polynômes abéliens de degré $\ell$. On s’intéresse ensuite aux ${\mu }_j$ associés à des éléments ${\theta }_u$ entiers, ce qui permet...