arXiv subjects
Publications and source records attributed to Jean-Luc Chabert.
In their study of the ring of integer-valued polynomials in non-commutative algebra, Peruginelli and Werner characterized the algebras for which this ring is a Pr\"ufer domain. Here, we apply their results to the case of group algebras.
The aim of this note is to give an elementary formulation of Tomio Kubota's proof of Satz 4 in his paper in Nagoya Math. t. 10 (1956) since this proposition is a cornerstone for the computation of the order of the Polya group of the biquadratic number fields.
The Polya group of a number field K is the subgroup of the class group of K generated by the classes of the products of the maximal ideals with same norm. A Polya field is a number field whose Polya group is trivial. Our purpose is to start with known assertions about Polya fields to find results concerning Polya groups. In this second paper we describe the Polya group of some non-galoisian extensions of Q.
To study the question of whether every two-dimensional Prüfer domain possesses the stacked bases property, we consider the particular case of the Prüfer domains formed by integer-valued polynomials. The description of the spectrum of the rings of integer-valued polynomials on a subset of a rank-one valuation domain enables us to prove that they all possess the stacked bases property. We also consider integer-valued polynomials on rings of integers of number fields and we reduce in this case the study of the stacked bases property to questions concerning $2\times 2$-matrices.
The Polya group of a number field K is the subgroup of the class group of K generated by the classes of the products of the maximal ideals with same norm. A Polya field is a number field whose Polya group is trivial. Our purpose is to start with known assertions about Polya fields to find results concerning Polya groups. This first paper inspired by Zantema's results focuses on the Polya group of the compositum of two galoisian extensions of Q.
We provide a simple proof of the fact that the ramification index of the compositum of two finite extensions of local fields is equal to the least common multiple of the ramification indices when at least one of the extensions is tamely ramified.
Let $\underline{E}=\prod_{p\in\mathbb{P}}E_p$ be a compact subset of $\widehat{\mathbb{Z}}=\prod_{p\in\mathbb{P}}\mathbb{Z}_p$ and denote by $\mathcal C(\underline{E},\widehat{\mathbb{Z}})$ the ring of continuous functions from $\underline{E}$ into $\widehat{\mathbb{Z}}$. We obtain two kinds of adelic versions of the Weierstrass approximation theorem. Firstly, we prove that the ring ${\rm Int}_{\mathbb{Q}}(\underline{E},\widehat{\mathbb{Z}}):=\{f(x)\in\mathbb{Q}[x]\mid \forall p\in\mathbb{P},\;\;f(E_p)\subseteq \mathbb{Z}_p\}$ is dense in the direct product $\prod_{p\in\mathbb{P}}\mathcal C(E_p,\mathbb{Z}_p)\,$ for the uniform convergence topology. Secondly, under the hypothesis that, for each $n\geq 0$, $\#(E_p\pmod{p})>n$ for all but finitely many $p$, we prove the existence of regular bases of the $\mathbb{Z}$-module ${\rm Int}_{\mathbb{Q}}(\underline{E},\widehat{\mathbb{Z}})$, and show that, for such a basis $\{f_n\}_{n\geq 0}$, every function $\underlineφ$ in $\prod_{p\in\mathbb{P}}\mathcal{C}(E_p,\mathbb{Z}_p)$ may be uniquely written as a series $\sum_{n\geq 0}\underline{c}_n f_n$ where $\underline{c}_n\in\widehat{\mathbb{Z}}$ and $\lim_{n\to \infty}\underline{c}_n\to 0$.
We show that every polynomial overring of the ring ${\rm Int}(\mathbb Z)$ of polynomials which are integer-valued over $\mathbb Z$ may be considered as the ring of polynomials which are integer-valued over some subset of $\hat{\mathbb{Z}}$, the profinite completion of $\mathbb Z$ with respect to the fundamental system of neighbourhoods of $0$ consisting of all non-zero ideals of $\mathbb Z.$