Galois module structure of some elementary \(p\)-abelian extensions
type: Item
instance of: scholarly article
title: Galois module structure of some elementary \(p\)-abelian extensions
author: Lauren Heller
author: Ján Mináč
author: Tung T. Nguyen
author: Andrew Schultz
author: Nguyên Duy Tân
published in: Israel Journal of Mathematics
publication date: 2024-08-08T00:00:00Z
review text: Let \(p\) be any prime. The set-up is as follows:\N\begin{itemize}\N\item \(F\) is field for which the maximal pro-\(p\) quotient of the absolute Galois group of \(F\) is a free and finitely generated pro-\(p\) group;\N\item \(F\) is assumed to contain a primitive \(p\)th root of unity when \(\mathrm{char}(F)\neq p\);\N\item \(K\) is a Galois extension of \(F\) for which \(\mathrm{Gal}(K/F)\) is a finite \(p\)-group.\N\end{itemize}\NConsider the parameterizing space \(J(K)\) of elementary \(p\)-abelian extensions of \(K\). By Artin-Schreier theory and Kummer theory, respectively, it is known that\N\[\NJ(K) = \begin{cases} K/\wp(K) & \mbox{if }\mathrm{char}(K)=p,\\\NK^\times/K^{\times p} &\mbox{if }\mathrm{char}(K)\neq p, \end{cases}\N\]\Nwhere \(\wp(K) = \{k^p - k :k\in K\}\) in the former case. Clearly \(J(K)\) is equipped with a natural \(\mathbb{F}_p\)-vector space structure. Denote by\N\[\N[F] = \begin{cases} (F+\wp(K))/\wp(K) & \mbox{if }\mathrm{char}(K)=p,\\\NFK^{\times p}/K^{\times p} &\mbox{if }\mathrm{char}(K)\neq p, \end{cases}\N\]\Nthe subspace of \(J(K)\) consisting of the elements which have a representative from \(F\).\N\NSince \(K/F\) is a Galois extension in the current context, one can also view \(J(K)\) as an \(\mathbb{F}_p[\mathrm{Gal}(K/F)]\)-module. In the case that \(\mathrm{Gal}(K/F)\) is a cyclic \(p\)-group, the module structure of \(J(K)\) has already been computed. In the paper under review, the authors allow \(\mathrm{Gal}(K/F)\) to be any finite \(p\)-group, and they were able to prove that\N\[\NJ(K) \simeq \Omega^{-2}_{\mathbb{F}_p[\mathrm{Gal}(K/F)]}(\mathbb{F}_p)\oplus Y,\N\]\Nwhere \( \Omega_{\mathbb{F}_p[\mathrm{Gal}(K/F)]}\) denotes the Heller operator and \(Y\) is a free module of rank \(\dim[F]\).
reviewed by: Cindy (Sin Yi) Tsang
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 16D10
zbMATH DE Number: 7893776
zbMATH Keywords: parameterizing space
zbMATH Keywords: Artin-Schreier theory
zbMATH Keywords: Kummer theory
zbMATH Keywords: Heller operator
zbMATH Keywords: Galois module structure
MaRDI profile type: Publication
cites work: Description of Galois unipotent extensions
cites work: The Sylow subgroups of the absolute Galois group \(\mathrm{Gal}(\mathbb{Q})\)
cites work: $p$-groups have unbounded realization multiplicity
cites work: The varieties and the cohomology ring of a module
cites work: Q4896043
cites work: Auslander-Reiten sequences for homotopists and arithmeticians
cites work: Galois \(p\)-groups and Galois modules
cites work: Galois module structure of square power classes for biquadratic extensions
cites work: On Galois groups over Pythagorean and semi-real closed fields
cites work: Q4849439
cites work: Field Arithmetic
cites work: Über modulare Darstellungen endlicher Gruppen, die von freien Gruppen induziert werden
cites work: Q4849452
cites work: On function fields with free absolute Galois groups
cites work: Q4553271
cites work: Q3663392
cites work: Q3726224
cites work: Newton's Identities Once Again!
cites work: Dimensions of Zassenhaus filtration subquotients of some pro-\(p\)-groups
cites work: GALOIS MODULE STRUCTURE OF pTH-POWER CLASSES OF CYCLIC EXTENSIONS OF DEGREE pn
cites work: Automatic realization of Galois groups with cyclic quotient of order \(p^n\)
cites work: Galois module structure of \(p\)th-power classes of extensions of degree \(p\)
cites work: Galois embedding problems with cyclic quotient of order \(p\)
cites work: Galois module structure of (ℓ n )th classes of fields
cites work: Étale Galois covers of affine smooth curves. The geometric case of a conjecture of Shafarevich. On Abhyankar's conjecture
cites work: Generic Galois extensions and problems in field theory
cites work: Parameterizing solutions to any Galois embedding problem over \(\mathbb{Z}/p^n\mathbb{Z}\) with elementary \(p\)-abelian kernel
cites work: Algebraic \(K\)-theory.
cites work: The Normal Closures of Certain Kummer Extensions
DOI: 10.1007/S11856-023-2540-6
description: scientific article; zbMATH DE number 7893776