On Vandiver's conjecture and \({\mathbb{Z}}_ p\)-extensions of \({\mathbb{Q}}(\zeta_{p^ n})\)
type: Item
instance of: scholarly article
title: On Vandiver's conjecture and \({\mathbb{Z}}_ p\)-extensions of \({\mathbb{Q}}(\zeta_{p^ n})\)
zbMATH Open document ID: 0709.11058
DOI: 10.1016/0022-314X(89)90091-7
author: Ina Kersten
author: Johannes Michaliček
published in: Journal of Number Theory
publication date: 1989-01-01T00:00:00Z
review text: Let \(K_ n={\mathbb{Q}}[\zeta_{n+1}]\), where \(\zeta_{n+1}\) is a primitive \(p^{n+1}\)-th root of unity, p an odd prime, and let \(K_ n^+\) be the maximal totally real subfield of K. Vandiver's conjecture is that p does not divide \(h_ 0^+\), the class number of \(K_ 0^+\). Let \(R_ n={\mathcal O}_{K_ n}[1/p]\); the authors relate Vandiver's conjecture to the structure of \({\mathbb{Z}}_ p\)-Galois extensions of \(R_ 0\), as follows. They first prove that if \(C_ n\) is the p-Sylow subgroup of the ideal class group of \(R_ n\), then the canonical map \(C_ n\to C_ m\) is injective for any \(m>n\) iff every \({\mathbb{Z}}_ p\)-extension of \(R_ n\) has normal basis. They use this to show: Vandiver's conjecture implies that each \({\mathbb{Z}}_ p\)-extension of \(R_ n\) has normal basis for all n, and conversely provided that the Iwasawa invariant \(\lambda^+\) of the cyclotomic \({\mathbb{Z}}_ p\)-extension of \(K_ 0^+\) is zero. Finally they prove that Vandiver's conjecture is equivalent to the property that each cyclic extension of \(K_ 0\) lying in \(H^-\) is contained in a \({\mathbb{Z}}_ p\)-extension of \(K_ 0\), where H is the Hilbert class field of \(K_ 0\) with Galois group \(G=G^+\oplus G^-\) and \(H^- =H^{G^+}\). These results strengthen the plausibility of Vandiver's conjecture because it is known from other work of the authors [J. Number Theory 32, 131-150 (1989; cf. the preceding review)] that in the group of \({\mathbb{Z}}_ p\)-extensions of \(R_ n\), the group of \({\mathbb{Z}}_ p\)- extensions with normal basis has finite index.
reviewed by: Lindsay N. Childs
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R23
zbMATH DE Number: 4164996
zbMATH Keywords: maximal totally real subfield
zbMATH Keywords: Vandiver's conjecture
zbMATH Keywords: class number
zbMATH Keywords: ideal class group
zbMATH Keywords: Iwasawa invariant
zbMATH Keywords: Hilbert class field
Wikidata QID: Q123196031
MaRDI profile type: Publication
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full work available at URL: 0022-314x(89)90091-7
OpenAlex ID: W2007681078
description: scientific article; zbMATH DE number 4164996