Generic Polynomials are Descent-Generic
نویسنده
چکیده
Let g(X) ∈ K(t1, . . . , tm)[X] be a generic polynomial for a group G in the sense that every Galois extension N/L of infinite fields with group G and K ≤ L is given by a specialization of g(X). We prove that then also every Galois extension whose group is a subgroup of G is given in this way. Let K be a field and G a finite group. Let us call a monic, separable polynomial g(t1, . . . , tm, X) ∈ K(t1, . . . , tm)[X] generic for G over K if the following two properties hold. (1) The Galois group of g (as a polynomial in X over K(t1, . . . , tm)) is G. (2) If L is an infinite field containing K and N/L is a Galois field extension with group G, then there exist λ1, . . . , λm ∈ L such that N is the splitting field of g(λ1, . . . , λm, X) over L. We call g descent-generic if it satisfies (1) and the stronger property (2’) If L is an infinite field containing K and N/L is a Galois field extension with group H ≤ G, then there exist λ1, . . . , λm ∈ L such that N is the splitting field of g(λ1, . . . , λm, X) over L. DeMeyer [2] proved that the existence of an irreducible descent-generic polynomial for a group G over an infinite field K is equivalent to the existence of a generic extension S/R for G over K in the sense of Saltman [6]. Ledet [5] proved that the existence of a generic polynomial for a group G over an infinite field K is equivalent to the existence of a generic extension S/R of G over K. Thus for K infinite the existence of a generic polynomial for G implies the existence of a descent-generic polynomial for G. In this note we prove the following stronger result. Theorem 1. Every generic polynomial g(t1, . . . , tm, X) for G over K is descent-generic. Proof. G has a faithful, transitive permutation representation G ↪→ Sn, by which it acts on the rational function field K(x1, . . . , xn). K(x1, . . . , xn) is Galois over K(x1, . . . , xn) with group G, hence there exist p1, . . . , pm ∈ K(x1, . . . , xn) such that K(x1, . . . , xn) is the splitting field of f(X) := g(p1, . . . , pm, X) over K(x1, . . . , xn). Write
منابع مشابه
Thematization Strategies in the Generic Moves of Research Article Introductions
Despite the heterogeneity of ideas regarding the definitions of genre, there are also common instances shared among scholars interested in particular aspects of the notion. Swales (1990) and Bhatia (1993) are primarily interested in the sociological and psychological aspects of genre's functioning and construction, respectively.. Swales analyzes the genre of 'article introduction', into four ge...
متن کاملDevelopment of a Generic Risk Matrix to Manage Project Risks
A generic risk matrix is presented for use identifying and assessing project risks quickly and cost effectively. It assists project managers with few resources to perform project risk analysis. The generic risk matrix (GRM) contains a broad set of risks that are categorized and ranked according to their potential impact and probability of occurrence. The matrix assists PMs in quickly identifyin...
متن کاملGeneric Extensions and Generic Polynomials for Semidirect Products
This paper presents a generalization of a theorem of Saltman on the existence of generic extensions with group A ⋊ G over an infinite field K, where A is abelian, using less restrictive requirements on A and G. The method is constructive, thereby allowing the explicit construction of generic polynomials for those groups, and it gives new bounds on the generic dimension. Generic polynomials for ...
متن کاملEffect generic and non-generic feedback on Motor Learning basketball free throw in Children
Non-generic feedback refers to a specific event and that task performance is the reason to the acquisition of skills and implies that performance is malleable, while generic feedback implies that task performance reflects an inherent ability. The Goal of this study was to determine the generic and non-generic feedback effects on children’s motor learning basketball free throw. This research was...
متن کاملThe Availability and Affordability of Cardiovascular Medicines for secondary prevention in Tehran Province
Abstract Abstract Background: Availability and affordability of medicines are crucial to achieving success in prevention programs, particularly in developing countries. Objective: The aim of this study was to determine the availability and affordability of cardiovascular medicines for secondary prevention in Tehran province of Iran. Methods: A cross-sectional survey was conducted in Tehran pro...
متن کامل