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Generic polynomial
About: Generic polynomial is a research topic. Over the lifetime, 608 publications have been published within this topic receiving 6784 citations.
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TL;DR: In this article, a polynomial for the cubic resolvent of a quartic with a rational point is given, whose Galois group is a subgroup of the 4-division field of an elliptic curve.
Abstract: There are 3 examples in these notes. The first one is the standard example of the cubic resolvent of a quartic. The second example is exactly from Adelmann \cite{Adelmann} and gives a defining polynomial corresponding to the unique $S_4$-quotient of $\mathrm{GL}_2(\mathbb{Z}/4\mathbb{Z})$. The splitting field of the Adelmann polynomial over $\mathbb{Q}$ is a subfield of the 4-division field of an elliptic curve, that contains the 2-division field of the elliptic curve. The third example is new and needed in the study of the field theory of quaternion origami. Associated to an elliptic curve defined over $\mathbb{Q}$, with a rational point, is a degree 8 polynomial whose Galois group is a subgroup of $\mathrm{Hol}(Q_8)$. Three defining polynomials corresponding to the three $S_4$-quotients of $\mathrm{Hol}(Q_8)$ are given.
1 citations
28 Aug 2006
TL;DR: In this paper, a deterministic polynomial-time algorithm was given to check whether the Galois group Gal(f) of an input polynomial f(X)∈ℚ[X] is nilpotent.
Abstract: We give a deterministic polynomial-time algorithm to check whether the Galois group Gal(f) of an input polynomial f(X)∈ℚ[X] is nilpotent: the running time is polynomial in size(f). Also, we generalize the Landau-Miller solvability test to an algorithm that tests if Gal(f) is in Γd: this algorithm runs in time polynomial in size(f) and nd and, moreover, if Gal(f)∈Γd it computes all the prime factors of # Gal(f).
1 citations
TL;DR: In this paper, the relation between Galois groups and Galois ensembles was described, where S is a submonoid of (the set of all natural numbers) and G is an associative group of inverse polynomial modules.
Abstract: Given an injective envelope E of a left R-module M, there is an associative Galois group Gal. Let R be a left noetherian ring and E be an injective envelope of M, then there is an injective envelope of an inverse polynomial module as a left R[x]-module and we can define an associative Galois group Gal. In this paper we describe the relations between Gal and Gal. Then we extend the Galois group of inverse polynomial module and can get Gal, where S is a submonoid of (the set of all natural numbers).
1 citations
01 Nov 2011
TL;DR: It is proved that any Galois connection on a complete lattice made an Armstrong system F(f, g) and the converse holds, that is, for a given Armstrong system R, the original Armstrong systems and Galois connections show us two faces of one thing.
Abstract: In the paper [1], it is proved that any Galois connection (f, g) on a complete lattice made an Armstrong system F (f, g) . We prove in this short note that the converse holds, that is, for a given Armstrong system R, we can make a Galois connection (φ R , ψ R ) and the original Armstrong system R is identical with the induced Armstrong system F (φR, ψR) by the Galois connection (φ R , ψ R ). This means that Armstrong systems and Galois connections show us two faces of one thing.
1 citations
TL;DR: For a given field F of characteristic 0, the authors considers a normal extension E/F of finite degree d and finite Abelian subgroups G⊂GLn(E) of a given exponent t and shows that any E can be realized as F(G), while if all coefficients of matrices in G are algebraic integers, there are only finitely many fields E=F(G) for prescribed integers n and t or prescribed n and d.
Abstract: For a given field F of characteristic 0 we consider a normal extension E/F of finite degree d and finite Abelian subgroups G⊂GLn(E) of a given exponent t. We assume that G is stable under the natural action of the Galois group of E/F and consider the fields E=F(G) that are obtained via adjoining all matrix coefficients of all matrices g∈G to F. It is proved that under some reasonable restrictions for n, any E can be realized as F(G), while if all coefficients of matrices in G are algebraic integers, there are only finitely many fields E=F(G) for prescribed integers n and t or prescribed n and d.
1 citations