Indépendance linéaire sur \=Q de logarithmes p-adiques de nombres algébriques et rang p-adique du groupe des unités d'un corps de nombres
type: Item
instance of: scholarly article
title: Indépendance linéaire sur \=Q de logarithmes p-adiques de nombres algébriques et rang p-adique du groupe des unités d'un corps de nombres
zbMATH Open document ID: 0547.12003
DOI: 10.1016/0022-314X(84)90079-9
author: H. H. Kisilevsky
author: Michel Emsalem
author: David Wales
published in: Journal of Number Theory
publication date: 1984-01-01T00:00:00Z
review text: Let k be a Galois extension of \({\mathbb{Q}}\) (resp. of an imaginary quadratic field \(k_ 0)\) and \(G=Gal(k/{\mathbb{Q}})\) (resp. \(G=Gal(k/k_ 0))\). Let E be the group of units of k, p a prime, and \(r_ p\) the p-adic rank of E. Finally denote by S the set of irreducible characters of G which occur in the decomposition of \(E\otimes {\bar {\mathbb{Q}}}\). Using Ax's method [\textit{J. Ax}, Ill. J. Math. 9, 584-589 (1965; Zbl 0132.283)], the authors deduce the lower bound \(r_ p=\sum_{\chi\in S}\deg \chi\) from the p- adic version of Baker's theorem on linear forms in logarithms [\textit{A. Brumer}, Mathematika 14, 121-124 (1967; Zbl 0171.011)]. As a corollary, they prove Leopoldt's conjecture for imaginary Galois extensions of \({\mathbb{Q}}\) with Galois group \(A_ 4\). They also give a lower bound for the p-adic rank of the p-units of k.
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11J81
Mathematics Subject Classification ID: 20C11
zbMATH DE Number: 3871495
zbMATH Keywords: p-adic logarithms of algebraic numbers
zbMATH Keywords: p-adic version of Baker's theorem
zbMATH Keywords: Leopoldt's conjecture
zbMATH Keywords: imaginary Galois extensions
zbMATH Keywords: Galois group \(A_ 4\)
zbMATH Keywords: lower bound for the p-adic rank of the p-units
reviewed by: Michel Waldschmidt
MaRDI profile type: Publication
cites work: On the units of an algebraic number field
cites work: Q3215038
cites work: On the units of algebraic number fields
cites work: Q4749922
cites work: Formulations de la conjecture de Leopoldt et étude d'une condition suffisante
cites work: On the maximal abelianℓ-extension of a finite algebraic number field with given ramification
full work available at URL: 0022-314x(84)90079-9
OpenAlex ID: W2027596513
description: scientific article; zbMATH DE number 3871495