Explicit Constructions of Generic Polynomials for Some Elementary Groups

Rikuna, Yūichi

doi:10.1007/978-1-4613-0249-0_9

Cited by 5 publications

(3 citation statements)

References 10 publications

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Section: Introductionmentioning

confidence: 99%

“…Note that we always take an infinite field M as a base field M, M ⊃ k, of a G-extension L/M. Examples of k-generic polynomials for G are known for various pairs of (k, G) (for example, see [Kem94], [KM00], [JLY02], [Rik04]). Let f G s (X) ∈ k(s)[X] be a k-generic polynomial for G. Kemper [Kem01] showed that for a subgroup H of G every H-Galois extension over an infinite field M ⊃ k is also given by a specialization of f G s (X) as in the similar manner.…”

Section: Introductionmentioning

confidence: 99%

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A Geometric Framework for the Subfield Problem of Generic Polynomials Via Tschirnhausen Transformation

Hoshi

Miyake

2009

Number Theory and Applications

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Geometric Framework for the Subfield Problem of Generic Polynomials Via Tschirnhausen Transformation

Hoshi

Miyake

2009

Number Theory and Applications

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“…[KM00], [JLY02], [Rik04]). Since a k-generic polynomial f G t (X) for G covers all G-Galois extensions over M ⊃ k by specializing parameters, it is natural to ask the following problem:…”

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confidence: 99%

On the Field Intersection Problem of Solvable Quintic Generic Polynomials

Hoshi

Miyake

2010

Int. J. Number Theory

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Let k be a field of characteristic = 2. We survey a general method of the field intersection problem of generic polynomials via formal Tschirnhausen transformation. We announce some of our recent results of cubic, quartic and quintic cases the details of which are to appear elsewhere. In this note, we give an explicit answer to the problem in the cases of cubic and dihedral quintic by using multi-resolvent polynomials. § 1. IntroductionLet G be a finite group, k a field of characteristic = 2, M a field containing k with #M = ∞, and k(t) the rational function field over k with n indeterminates t = (t 1 , . . . , t n ). Our main interest in this note is a k-generic polynomial for G (cf.[DeM83], [Kem01], [JLY02]).Definition. A polynomial f t (X) ∈ k(t)[X] is called k-generic for G if it has the following property: the Galois group of f t (X) over k(t) is isomorphic to G and every G-Galois extension L/M over an arbitrary infinite field M ⊃ k can be obtained as L = Spl M f a (X), the splitting field of f a (X) over M , for some a = (a 1 , . . . , a n ) ∈ M n .be a k-generic polynomial for G. Examples of k-generic polynomials for G are known for various pairs of (k, G) (for example, see [Kem94],

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Noether's problem and $\mathbb{Q}$-generic polynomials for the normalizer of the $8$-cycle in $S_8$ and its subgroups

2007

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Abstract. We study Noether's problem for various subgroups H of the normalizer of a group C 8 generated by an 8-cycle in S 8 , the symmetric group of degree 8, in three aspects according to the way they act on rational function fields, i.e., Q(X 0 , . . . , X 7 ), Q(x 1 , . . . , x 4 ), and Q(x, y). We prove that it has affirmative answers for those H containing C 8 properly and derive a Q-generic polynomial with four parameters for each H. On the other hand, it is known in connection to the negative answer to the same problem for C 8 /Q that there does not exist a Q-generic polynomial for C 8 . This leads us to the question whether and how one can describe, for a given field K of characteristic zero, the set of C 8 -extensions L/K. One of the main results of this paper gives an answer to this question.

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Explicit Constructions of Generic Polynomials for Some Elementary Groups

Cited by 5 publications

References 10 publications

A Geometric Framework for the Subfield Problem of Generic Polynomials Via Tschirnhausen Transformation

A Geometric Framework for the Subfield Problem of Generic Polynomials Via Tschirnhausen Transformation

On the Field Intersection Problem of Solvable Quintic Generic Polynomials

Noether's problem and $\mathbb{Q}$-generic polynomials for the normalizer of the $8$-cycle in $S_8$ and its subgroups

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