On the Leopoldt conjecture on the p-adic regulators
type: Item
instance of: scholarly article
title: On the Leopoldt conjecture on the p-adic regulators
zbMATH Open document ID: 0621.12009
DOI: 10.1016/0022-314X(87)90073-4
author: Hiroo Miki
published in: Journal of Number Theory
publication date: 1987-01-01T00:00:00Z
review text: Leopoldt's conjecture predicting the non-vanishing of the p-adic regulator \(R_ p(k)\) of an algebraic number field k has been proven by Ax and Brumer in case k is an abelian extension of \({\mathbb{Q}}\) or of any imaginary quadratic number field, using an analytic method of Baker. In the present paper the author investigates the problem how to conclude purely algebraically the non-vanishing of \(R_ p(K)\) for a cyclic extension K/k of degree p under the assumption \(R_ p(k)\neq 0\). For \(k={\mathbb{Q}}\) or k imaginary quadratic this is carried out successfully and moreover an explicite lower bound for \(| R_ p(K)|_ p\) is given for \(k={\mathbb{Q}}\). The conclusion is also proven in a more general situation under certain strong conditions on k and on the ramification of K/k. Furthermore the Galois group \(G_ k(p)\) of the maximal p-ramified abelian p-extension of k is taken into consideration. For a Galois extension K/k of p-power degree a condition is worked out which from \(R_ p(k)\neq 0\) and \(G_ k(p)\) torsion free concludes \(R_ p(K)\neq 0\) and \(G_ K(p)\) torsion free.
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11R34
zbMATH DE Number: 4006375
zbMATH Keywords: Leopoldt's conjecture
zbMATH Keywords: p-adic regulator
zbMATH Keywords: Galois group
zbMATH Keywords: maximal p-ramified abelian p-extension
reviewed by: Claus Günther Schmidt
MaRDI profile type: Publication
Wikidata QID: Q123009471
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description: scientific article; zbMATH DE number 4006375