منابع مشابه
Generic Polynomials are Descent-Generic
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) ...
متن کاملGeneric Polynomials with Few Parameters
We call a polynomial g(t1, . . . , tm, X) over a field K generic for a group G if it has Galois group G as a polynomial in X, and if every Galois field extension N/L with K ⊆ L and Gal(N/L) ≤ G arises as the splitting field of a suitable specialization g(λ1, . . . , λm, X) with λi ∈ L. We discuss how the rationality of the invariant field of a faithful linear representation leads to a generic p...
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 کاملComputing Galois Groups with Generic Resolvent Polynomials
Given an arbitrary irreducible polynomial f with rational coefficients it is difficult to determine the Galois group of the splitting field of that polynomial. When the roots of f are easy to calculate, there are a number of “tricks” that can be employed to calculate this Galois group. If the roots of f are solvable by radicals, for example, it is often easy to calculate by hand the Q-fixing au...
متن کاملA Family of Polynomials Related to Generic Degrees
A family of polynomials parameterized by the conjugacy classes of a finite Coxeter group is investigated. These polynomials, together with the character table of the group, determine the associated generic degrees. The polynomials are described completely for classes that meet a parabolic subgroup whose components are of type A or are dihedral, and for the class of Coxeter elements.
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ژورنال
عنوان ژورنال: Polythematic Online Scientific Journal of Kuban State Agrarian University
سال: 2017
ISSN: 1990-4665
DOI: 10.21515/1990-4665-132-044