Correspondances compatibles avec une relation binaire, relevement d'extensions de groupe de Galois L3(2) et probleme de Noether pour L3(2)

2004/02/11 by J. -F. Mestre, Jean-François Mestre, Mestre, J. -F.
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT) #math.GR #math.NT

paper · pdf · doi:10.48550/arxiv.math/0402187

The paper is partially re-written, and a section where we prove that the Noether's problem is true for L3(2) is added

openalex publication_date 2004/02/11 · arxiv created 2005/08/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that for a system of indeterminates (X_a) indiced by the P²(2), the projective plane over F₂, there exists a 3-3 correspondance compatible with the incidence structures of P²(2), such that (X_a) is one of the orbits of it. We give two applications of this construction : 1) for any sufficientely general polynomial P in k[X] over a field k of car. 0, such that its Galois group is a subgroup of L3(2) ((=L2(7)), there exists Q in k[X] such that the Galois group of P-TQ over k(T) is L3(2). This implies in particular the so-called "arithmetical lifting property" for L3(2) over k. 2) There exists a generic polynomial in 7 parameters for polynomials of degree 7 with Galois group L3(2). This is equivalent to the fact that the Noether's problem for L3(2) acting over the seven points of P²(2) has a positive answer.

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