Finite \(p\)-groups and \(k((t))\).
type: Item
instance of: scholarly article
title: Finite \(p\)-groups and \(k((t))\).
zbMATH Open document ID: 1251.20022
author: Anthony J. Bevelacqua
author: Mark Motley
published in: Journal of Algebra
publication date: 2012-03-22T00:00:00Z
review text: Let \(p\) be a prime, \(G\) a finite group of order \(p^n\), \(k\) a field of characteristic \(p>0\) and \(K=k((t))\) the field of finitely tailed Laurent series over \(k\). Then there exists a unit \(u\in kt\) such that \(K/k((ut^{p^n}))\) is Galois with Galois group isomorphic to \(G\). The authors investigate the relationship between \(u\) and \(G\). In particular, they prove that if \(u_1\) and \(u_2\) are units that are sufficiently close in the \(t\)-adic topology then \(\text{Gal}(K/L_1)\cong\text{Gal}(K/L_2)\) where \(L_i=k((u_it^{p^n}))\) for \(i=1,2\). The authors raise interesting questions, asking whether properties of the coefficients of \(u\) determine algebraic properties of \(G\), for example whether \(G\) is cyclic or Abelian. Conversely, do properties of \(G\) determine the sequence of coefficients of \(u\)? Note, when \(k\) is a finite field these investigations relate to results about finite \(p\)-groups embedding in the so-called Nottingham group.
reviewed by: Rachel D. Camina
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 12F10
zbMATH DE Number: 6017051
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: Laurent series
zbMATH Keywords: Nottingham group
zbMATH Keywords: Galois groups
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2011.08.002
OpenAlex ID: W1562506426
cites work: Subgroups of the Nottingham group
cites work: Q3818488
cites work: Finite Codimension Subfields of a Field Complete with Respect to a Real Valuation
cites work: Profinite Groups, Arithmetic, and Geometry. (AM-67)
cites work: Q4950147
DOI: 10.1016/J.JALGEBRA.2011.08.002
description: scientific article; zbMATH DE number 6017051