Cyclic Galois extensions of commutative rings
type: Item
instance of: scholarly article
title: Cyclic Galois extensions of commutative rings
zbMATH Open document ID: 0788.13003
author: Cornelius Greither
published in: Lecture Notes in Mathematics
publication date: 1993-11-22T00:00:00Z
review text: Let \(R\) be a commutative ring, \(G\) a finite abelian group. The group \(\mathrm{Gal}(R,G)\) of isomorphism classes of Galois extensions of \(R\) with group \(G\), as developed by Auslander, Goldman, Chase, Harrison and Rosenberg in the 1960s, has important connections with algebraic number theory: if \(R\) is the ring of integers of a number field \(K\), then by class field theory \(\mathrm{Gal}(R,G)\) is closely related to the class group of \(K\); more recently, work of Kersten, Michalichek and the author has found important and subtle connections with Leopoldt's and Vandiver's conjectures [c.f. \textit{I. Kersten} and \textit{J. Michalichek} [J. Number Theory 32, 371--386 (1989; Zbl 0709.11058)]. The main purpose of these notes is to describe \(\mathrm{Gal}(R,G)\) in a noncohomological way. Since \(\mathrm{Gal}(R,G)\) respects products in the second variable, the problem immediately reduces to \(G\) cyclic of prime power order \(p^n\), which we assume henceforth. Chapter 0 gives a useful description of the basic theory and a description of \(\mathrm{Gal}(R,G)\) and the subgroup \(\mathrm{NB}(R,G)\) of Galois extensions with normal basis, when \(R\) is connected and contains \(p^{-1}\) and a primitive \(p^n\)-th root of unity \(\zeta\). This generalized Kummer theory goes back to the reviewer [Ill. J. Math. 15, 273--280 (1971; Zbl 0211.37102)] and \textit{A. Z. Borevich} [J. Sov. Math. 11, 514--534 (1979); transl. from Zap. Nauchn. Semin. LOMI Steklova 57, 8--30 (1976; Zbl 0379.13003)]. Chapter I then describes \(\mathrm{Gal}(R,G)\) when \(R\) contains \(p^{-1}\) but not \(\zeta\), by letting \(S=R[\zeta]\) and ``descending'' the Kummer theory over \(S\) to \(R\). This chapter is based on the author's paper [Trans. Am. Math. Soc. 326, 307--343 (1991; Zbl 0743.11060)], and has useful applications to number theory, as is shown in chapter IV. Chapter II describes an alternate approach to obtaining \(\mathrm{Gal}(R,G)\) from \(\mathrm{Gal}(S,G)\) using corestrictio…
Mathematics Subject Classification ID: 13B05
Mathematics Subject Classification ID: 13-02
Mathematics Subject Classification ID: 11-02
Mathematics Subject Classification ID: 13B02
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11S15
Mathematics Subject Classification ID: 14E22
Mathematics Subject Classification ID: 14E20
zbMATH DE Number: 462628
zbMATH Keywords: extensions of function field
zbMATH Keywords: generic Galois extension
zbMATH Keywords: Kummer theory
zbMATH Keywords: Leopoldt's conjecture
zbMATH Keywords: cyclotomic fields
zbMATH Keywords: geometric class field theory
reviewed by: Lindsay N. Childs
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0089165
OpenAlex ID: W1558411995
DOI: 10.1007/BFB0089165
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description: scientific article; zbMATH DE number 462628