[過去ﾛｸﾞ] 現代数学の系譜 工学物理雑談 古典ガロア理論も読む78 (1002ﾚｽ)
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494(2): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/04(月)19:56 ID:Qu1TcOyQ(16/28) AAS
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これ、結構 Inverse Galois Problem でよく纏まっているね
外部ﾘﾝｸ[pdf]:arxiv.org
Inverse Galois Problem and Significant Methods
Fariba Ranjbar*
, Saeed Ranjbar
* School of Mathematics, Statistics and Computer Science, University of Tehran,
Tehran, Iran.
(抜粋)
ABSTRACT. The inverse problem of Galois Theory was developed in the early
1800’s as an approach to understand polynomials and their roots. The inverse
Galois problem states whether any finite group can be realized as a Galois
group over ? (field of rational numbers). There has been considerable progress
in this as yet unsolved problem. Here, we shall discuss some of the most
significant results on this problem. This paper also presents a nice variety of
significant methods in connection with the problem such as the Hilbert
irreducibility theorem, Noether’s problem, and rigidity method and so on.
I. Introduction
Is every finite group realizable as the Galois group of a Galois extension of ??
(A) General existence problem. Determine whether G occurs as a Galois group over
K. In other words, determine whether there exists a Galois extension M/K such that
the Galois group Gal (M/K) is isomorphic to G. We call such a Galois extension M a
G-extension over K.
(B) Actual construction. If G is realisable as a Galois group over K, construct
explicit polynomials over K having G as a Galois group. More generally, construct a
family of polynomials over a K having G as Galois group.
Is every finite group realizable as the Galois group of a Galois extension of Q?
II. Milestones in Inverse Galois Theory
For the 26 sporadic simple groups, all but possibly one, namely, the Mathieu group
M23, have been shown to occur as Galois groups over Q.
495: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/04(月)19:58 ID:Qu1TcOyQ(17/28) AAS
>>494 関連
Mathieu group M23
”The inverse Galois problem seems to be unsolved for M23. In other words, no polynomial in Z[x] seems to be known to have M23 as its Galois group. The inverse Galois problem is solved for all other sporadic simple groups.”
なのか
へー
外部ﾘﾝｸ:en.wikipedia.org
Mathieu group M23
(抜粋)
History and properties
Milgram (2000) calculated the integral cohomology, and showed in particular that M23 has the unusual property that the first 4 integral homology groups all vanish.
The inverse Galois problem seems to be unsolved for M23. In other words, no polynomial in Z[x] seems to be known to have M23 as its Galois group. The inverse Galois problem is solved for all other sporadic simple groups.
498(2): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/04(月)20:36 ID:Qu1TcOyQ(19/28) AAS
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外部ﾘﾝｸ:www.sciencedirect.com
Journal of Algebra
Volume 422, 15 January 2015, Pages 187-222
Parametric Galois extensions
Author links open overlay panelFrancoisLegrand
外部ﾘﾝｸ:reader.elsevier.com
外部ﾘﾝｸ:www.researchgate.net
The Inverse Galois Problem (4th year project).
Article (PDF Available) ・ May 2017
Dean Yates
Queen Mary, University of London
Abstract
For a given finite group G, the 'Inverse Galois Problem' consists of determining whether G occurs as a Galois group over a base field K,
or in other words, determining the existence of a Galois extension L of the base field K such that G is isomorphic to the group of automorphisms on L
(under the group operation of composition) that fix the elements of K. Having established the existence of such a field extension, with a specified group G as its Galois group,
one then seeks to construct an explicit family of polynomials over K having G as its Galois group.
We focus in particular on the classical problem, where our base field K is the field of rational numbers, and explore the classical problem for a variety of finite groups.
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