Stickelberger's criterion, Galois algebras, and tame ramification in algebraic number fields
type: Item
instance of: scholarly article
title: Stickelberger's criterion, Galois algebras, and tame ramification in algebraic number fields
zbMATH Open document ID: 0558.12001
DOI: 10.1016/0022-4049(84)90063-X
author: Donald Maurer
published in: Journal of Pure and Applied Algebra
publication date: 1984-01-01T00:00:00Z
review text: Let F be an algebraic number field in which the odd prime p is unramified. For a cyclic Galois extension K of F of degree p, let \({\mathcal S}_{K/F}\) be the set of primes of F which ramify tamely in K. The main theorem of this paper, generalizing work of \textit{S. Ullom} [J. Number Theory 6, 32-49 (1974; Zbl 0278.12004)], is a characterization of finite sets \({\mathcal S}=\{q_ 1,...,q_ m\}\) of prime ideals of F such that \({\mathcal S}={\mathcal S}_{K/F}\) for some K. The characterization involves the known necessary condition that each prime q in \({\mathcal S}\) splits completely in F(\(\zeta)\), \(\zeta\) a primitive pth root of unity. If \(Q_ i\) is a prime divisor of \(q_ i\) in F(\(\zeta)\), and \({\mathcal A}=\prod^{m}_{i=1}Q_ i\), then the characterization \({\mathcal S}={\mathcal S}_{K/F}\) is given in terms of a congruence condition on the images cl \({\mathcal A}^{\tau (m)}\) of the ideal class of \({\mathcal A}\) under the action of Stickelberger elements \(\tau\) (m) in \({\mathbb{Q}}\Delta\), \(\Delta =Gal({\mathbb{Q}}(\zeta)/{\mathbb{Q}})\). The author characterizes in a similar way those \({\mathcal S}\) with \({\mathcal S}={\mathcal S}_{K/F}\) for K tamely ramified or with normal integral basis. To obtain these results, the author first gives an arithmetic classification of the set of Galois R-algebras with Galois group G, cyclic of prime order p, when R is a commutative ring with 2p a non-zero divisor. Here a Galois R-algebra A is in the sense of Hasse: \(A=RGa\cong RG\) as RG-module. The classification is obtained by letting S be the cyclotomic extension obtained by adjoining a formal pth root of unity to R, computing the resolvant (a\(| \chi)\) in \(A\otimes S\) for \(\chi\) a generator of the character group (into S) of G, and associating to A the element \(\theta =(a| \chi)^ p/tr(a)\) in \(A\otimes S\). The resolvant powers \(\theta\) arising in this way are characterized by congruence conditions involving the images of \(\theta\) under th…
reviewed by: Lindsay N. Childs
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 13B05
Mathematics Subject Classification ID: 11S15
Mathematics Subject Classification ID: 11R52
zbMATH DE Number: 3889660
zbMATH Keywords: cyclic Galois extension
zbMATH Keywords: Stickelberger elements
zbMATH Keywords: tamely ramified
zbMATH Keywords: normal integral basis
zbMATH Keywords: arithmetic classification
zbMATH Keywords: Galois R-algebras
zbMATH Keywords: cyclotomic extension
zbMATH Keywords: resolvant
zbMATH Keywords: congruence conditions
zbMATH Keywords: Harrison group
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3889660