現代数学の系譜工学物理雑談古典ガロア理論も読む78

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498: 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) 20:36:33.27 ID:Qu1TcOyQ

>>494 追加

https://www.sciencedirect.com/science/article/pii/S0021869314005183
Journal of Algebra
Volume 422, 15 January 2015, Pages 187-222
Parametric Galois extensions
Author links open overlay panelFrancoisLegrand
https://reader.elsevier.com/reader/sd/pii/S0021869314005183?token=4C069AE76EFFCFC1E18B0BFD69A31AF06C2CC66DB402642D923675313C2A41FBF9CF61E485C1544038585012634EFB5B
https://www.researchgate.net/publication/320835842_The_Inverse_Galois_Problem_4th_year_project
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.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1571400076/498

499: 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) 20:38:05.98 ID:Qu1TcOyQ

>>498

つづき

Download full-text PDF
https://www.researchgate.net/profile/Dean_Yates/publication/320835842_The_Inverse_Galois_Problem_4th_year_project/links/59fc9ddf458515d070654852/The-Inverse-Galois-Problem-4th-year-project.pdf
(抜粋)
6. Table of finite groups realisable over Q
We will now bring together results from the preceding sections to form a table of finite
groups G that can be realised as Galois groups of rational polynomials and their corresponding
generic polynomials: [16]
G G-polynomial over Q generic polynomial for G
V4 (t^4 + 1) (t^2 - x)(t^2 - y)
Cn Φn(t) exist iff. 8 - n
Sn -15f1 + 10f2 + 6f3 (c.f. theorem 3.3) t^n + an-1t^n-1 + + a1t + a0
S4 t^4 + 16t^3 - 4t^2 + 3t - 11 t^4 + xt^2 + yt + y
An p(x; t) (c.f. theorem 4.5) unknown
A3 t^3 - 3t + 1 t^3 - xt^2 + (x - 3)t + 1
A4 t^4 - 2t^3 + 2t^2 + 2 F((x; y); t) (see below)
Dn unknown in general exist iff. 4 - n
D8 t^4 - 2 t^4 - 4xt^2 + y
D10 t^5 - 5t^2 - 3 t^5 + (y - 3)t^4 + (x - y + 3)t^3+(y2 - y - 2x - 1)t^2 + xt + y
(引用終り)

http://rio2016.5ch.net/test/read.cgi/math/1571400076/499

504: 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) 20:54:59.90 ID:Qu1TcOyQ

>>498 追加

”Shafarevich's theorem on solvable Galois groups”
https://en.wikipedia.org/wiki/Shafarevich%27s_theorem_on_solvable_Galois_groups
Shafarevich's theorem on solvable Galois groups

In mathematics, Shafarevich's theorem states that any finite solvable group is the Galois group of some finite extension of the rational numbers. It was first proved by Igor Shafarevich (1954), though Schmidt[who?] later pointed out a gap in the proof, which was fixed by Shafarevich (1989).

https://arxiv.org/abs/math/9809211
Safarevic's theorem on solvable groups as Galois groups
Alexander Schmidt, Kay Wingberg
(Submitted on 17 Sep 1998)
https://arxiv.org/pdf/math/9809211.pdf

The aim of this article is to give a complete proof of the following famous
theorem of I. R. Safarevic:
Theorem 1
Let k be a global field and let G be a finite solvable group.
Then there exists a finite Galois extension K｜k with Galois group G(K｜k) ～= G.

http://rio2016.5ch.net/test/read.cgi/math/1571400076/504

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