Kummer theory without roots of unity
type: Item
instance of: scholarly article
title: Kummer theory without roots of unity
zbMATH Open document ID: 0638.13007
DOI: 10.1016/0022-4049(88)90003-5
published in: Journal of Pure and Applied Algebra
publication date: 1988-01-01T00:00:00Z
review text: Let \(R\) be a connected commutative ring with group of units \(R^*\), \(p\) an odd prime which is a unit in \(R\), \(\Gamma_n\) a group cyclic of order \(q=p^n\) with generator \(\tau_n\), \(G_n\) its dual with generator \(\sigma_n\). The authors classify the group \(H(R,\Gamma_n)\) of Galois extensions of \(R\) with group \(\Gamma_n\) having normal basis (c.f. \textit{S. U. Chase}, \textit{D. K. Harrison} and \textit{A. Rosenberg} [Mem. Am. Math. Soc. 52, 15--33 (1965; Zbl 0143.05902)]); when \(R\) contains a primitive \(q\)-th root of unity, it is well-known that the group \(H(R,\Gamma_n)\) is isomorphic to \(R^*/R^{*q}\) by classical Kummer theory. The authors do not assume \(R\) contains roots of unity, and rather than adjoining a \(q\)-th root of unity, using the classical theory and then descending (\textit{C. Greither}'s approach in his 1988 München Habilitationsschrift, written after this paper) they associate to a normal basis of a \(\Gamma_n\)-Galois extension an element of the group ring \(R[G_n]\), and then identify the resulting classes of elements. More precisely, they define a subgroup \(C_n(R)\) of \(R(G_n)^*\), define a homomorphism \(\phi\) from \(C_n(R)\) to \(H(R,\Gamma_n)\), and identify the kernel of \(\phi\). To accomplish this, the authors first show that if \(A\) is a \(\Gamma_n\)-Galois extension of \(R\), then \(X_0,\dots,X_{q-1}\) forms a normal basis of \(A\) iff \(X=\sum_{i=0}^{q-1}X_i\sigma_n^{-i}\) is a unit of \(A[G_n]\) and \(\tau_n(X)=\sigma_n(X)\). They call such an \(X\) in \(A[G_n]\) a normal basis of \(A\) over \(R\). A normal basis \(X\) of \(A\) over \(R\) can be found which is normalized: if \(v_k: A[G_n]\to A[G_n]\) is induced by sending \(\sigma\) to \(\sigma^k\), then \(X\) in \(A[G_n]^*\) is normalized if \(v_0(X)= \mathrm{tr}(X_0)=1\) and \(v_{-1}(X)X=1\). Let \(q\), \(w\) be the multiplicative maps \(A[G_n]^*\to A[G_n]^*\) defined by \(q(x)=x^{p+1}/v_{p+1}(x)\); \(w(x)=x^p/v_p(x)\). Then \(q\) and \(w\) commute. If \(…
Mathematics Subject Classification ID: 13B05
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 12G05
zbMATH DE Number: 4040046
zbMATH Keywords: Kummer theory
zbMATH Keywords: Galois extension
zbMATH Keywords: normal basis
author: Ina Kersten
author: Johannes Michaliček
reviewed by: Lindsay N. Childs
MaRDI profile type: Publication
full work available at URL: 0022-4049(88)90003-5
OpenAlex ID: W1963663418
cites work: Q5518494
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cites work: A Cohomological Description of Abelian Galois Extensions
cites work: Generic Galois extensions and problems in field theory
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description: scientific article; zbMATH DE number 4040046