Prehomogeneous vector spaces and field extensions
type: Item
instance of: scholarly article
title: Prehomogeneous vector spaces and field extensions
zbMATH Open document ID: 0803.12004
DOI: 10.1007/BF01231334
author: David J. Wright
author: Akihiko Yukie
published in: Inventiones Mathematicae
publication date: 1993-05-16T00:00:00Z
full work available at URL: https://eudml.org/doc/144052
review text: The purpose of this paper is to give a correspondence between the set of orbits of certain prehomogeneous vector spaces and the set of isomorphism classes of Galois extensions of a field \(k\). Let \(k\) be an infinite field of characteristic \(\neq 2,3,5\). The set \({\mathfrak {Gr}}_ i\) is, by definition, the set of isomorphism classes of Galois extensions of \(k\) which are splitting fields of degree \(i\) equations without multiple roots. Let \((G_ k, V_ k)\) be a prehomogeneous vector space with a relative invariant \(\Delta(x)\). We set \(V_ k^{ss}:= \{x\in V_ k\); \(\Delta(x)\neq 0\}\). For \(2\leq i\leq 5\), the authors construct a natural map \(\alpha_ V: G_ k \setminus V_ k^{ss}\ni x\mapsto k(x)\in {\mathfrak {Gr}}_ i\) for certain prehomogeneous vector space \((G_ k, V_ k)\). For each point \(x\in V_ k^{ss}\), they associate a homomorphism \(p_ k: \text{Gal} (k^{\text{sep}}/k) \to{\mathfrak S}_ i\) where \(\text{Gal}(-)\) stands for the Galois group of separable extensions over \(k\) and \({\mathfrak S}_ i\) is the permutation group of degree \(i\). The main result of this paper is the following: let \(x,y\in V_ k^{ss}\) and \(k(x)= k(y)= k'\in {\mathfrak {Gr}}_ i\). Then \(x\) and \(y\) are \(G_ k\)- equivalent if and only if there exists \(r\in{\mathfrak S}\) such that \(p_ x= rp_ y r^{-1}\).
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 11S90
Mathematics Subject Classification ID: 14M17
zbMATH DE Number: 167396
zbMATH Keywords: orbits of prehomogeneous vector spaces
zbMATH Keywords: isomorphism classes of Galois extensions
zbMATH Keywords: infinite field
zbMATH Keywords: splitting fields
zbMATH Keywords: separable extensions
reviewed by: Masakazu Muro
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 167396