現代数学の系譜工学物理雑談古典ガロア理論も読む78

[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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462: 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) 17:47:37.21 ID:Qu1TcOyQ

>>421
>PSL(2,16):2 が、なにか、ガロア逆問題から見て、特別な存在なのでしょうね

ちょっと検索でヒットしたので貼る（＾＾
この ”Multi-parameter polynomials”が面白いと思った
例えば、”5. The group PGL2(7)”は、ガロア逆問題は解けているみたい

https://www.mathematik.uni-kl.de/agag/personen/leitung/
Technische Universitat Kaiserslautern
https://www.mathematik.uni-kl.de/~malle/de/publications.html
Prof. Dr. Gunter Malle Veroffentlichungen
https://link.springer.com/book/10.1007/978-3-662-55420-3
136 (mit B. H. Matzat): Inverse Galois Theory. 2nd Edition. Springer Verlag (2018), xvii + 533 pp., (MR3822366).
https://www.mathematik.uni-kl.de/~malle/download/pargal.pdf
55 Multi-parameter polynomials with given Galois group. J. Symbolic Comput. 30 (2000), 717-731, (MR 2002a:12007).
GUNTER MALLE FB Mathematik, Universit¨at Kassel, Heinrich-Plett-Str. 40, 34132 Kassel, Germany
(抜粋)
We present a collection of multi-parameter polynomials for several mostly non-solvable
permutation groups of small degree and describe their construction. As an application we
are able to obtain totally real number fields with these Galois groups over the rationals,
for example for the two small Mathieu groups M11 and M12.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1571400076/462

463: 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) 17:47:58.44 ID:Qu1TcOyQ

>>462
つづき

1. Introduction
The non-abelian finite simple groups and their automorphism groups play a crucial role
in an inductive approach to the inverse problem of Galois theory. The rigidity method
(see for example Malle and Matzat (1999)) has proved very efficient for deducing the
existence of Galois extensions with such groups, as well as for the construction of polynomials generating such extensions. Nevertheless, the effective construction requires the
solution of a non-linear system of equations, a problem which is known to be very hard
from a complexity point of view. Thus, in practice, the computation of polynomials is
restricted to rather small degree, to the case of stem fields of genus zero and also to few
(mostly three) ramification points. For several applications, for example for the solution
of embedding problems, it is sometimes necessary to find Galois extensions of the rationals with given group and with complex conjugation lying in a prescribed conjugacy
class. But it is well known (see for example Malle and Matzat (1999), Ex. I.10.2) that
three point ramified Galois extensions almost never have totally real specializations, for
example.
In this paper we give 2-, 3- and 4-parameter polynomials for certain (mostly nonsolvable) groups which, from a certain point of view, correspond to Galois extensions
ramified in at least four points, with the property that these admit (infinitely many) totally real, Galois group preserving specializations. For example we obtain a two-parameter
polynomial for the sporadic simple Mathieu group M12 over Q. Suitable specializations
then yield totally real number fields with groups M11 and M12.
Acknowledgement: I would like to thank Peter M¨uller for very useful conversations on
the topic of this paper.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1571400076/463

466: 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) 18:03:34.00 ID:Qu1TcOyQ

>>462 追加

”PARAMETRIC GALOIS EXTENSIONS”
https://arxiv.org/pdf/1310.6682.pdf
PARAMETRIC GALOIS EXTENSIONS
FRANC， OIS LEGRAND
Abstract. Given a field k and a finite group H, an H-parametric
extension over k is a finite Galois extension of k(T ) of Galois group
containing H which is regular over k and has all the Galois extensions of k of group H among its specializations. We are mainly interested in producing non H-parametric extensions, which relates
to classical questions in inverse Galois theory like the BeckmannBlack problem and the existence of one parameter generic polynomials. We develop a general approach started in a preceding paper
and provide new non parametricity criteria and new examples.

1. Presentation
The Inverse Galois Problem asks whether, for a given finite group H,
there exists at least one Galois extension of Q of group H. A classical
way to obtain such an extension consists in producing a Galois extension E/Q(T) with the same group which is regular over Q 1
: from the Hilbert irreducibility theorem, E/Q(T) has at least one specialization
of group H (in fact infinitely many if H is not trivial).
In this paper we are interested in “parametric Galois extensions”,
i.e. in finite Galois extensions E/Q(T) which are regular over Q - from
now on, say for short that E/Q(T) is a “Q-regular Galois extension”
- and which have all the Galois extensions of Q of group H among
their specializations. More precisely, given a field k and a finite group
H, we say that a k-regular finite Galois extension E/k(T) of group G
containing H (with possibly H 6= G) is H-parametric over k if any
Galois extension of k of group H ocurs as a specialization of E/k(T)
(definition 2.2). The special case H = G is of particular interest.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1571400076/466

494: 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) 19:56:43.30 ID:Qu1TcOyQ

>>462 追加

これ、結構 Inverse Galois Problem でよく纏まっているね
https://arxiv.org/ftp/arxiv/papers/1512/1512.08708.pdf
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.

http://rio2016.5ch.net/test/read.cgi/math/1571400076/494

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