現代数学の系譜工学物理雑談古典ガロア理論も読む77

[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77 (1002ﾚｽ)
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961: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/10/18(金) 07:18:10.93 ID:Zm+yHrIo

>>918
>・B_{5}'メタ巡回群 (位数 20)

追加参考
（後の”Metacyclic Group Wolfram MathWorld”の方が、綺麗に纏まっているが、書きぶりがちょっと違う）
https://en.wikipedia.org/wiki/Metacyclic_group
Metacyclic group
(抜粋)
In group theory, a metacyclic group is an extension of a cyclic group by a cyclic group. That is, it is a group G for which there is a short exact sequence
1 → K → G → H → 1
where H and K are cyclic. Equivalently, a metacyclic group is a group G having a cyclic normal subgroup N, such that the quotient G/N is also cyclic.
Properties
Metacyclic groups are both supersolvable and metabelian.
Examples
・Any cyclic group is metacyclic.
・The direct product or semidirect product of two cyclic groups is metacyclic. These include the dihedral groups and the quasidihedral groups.
・The dicyclic groups are metacyclic. (Note that a dicyclic group is not necessarily a semidirect product of two cyclic groups.)
・Every finite group of squarefree order is metacyclic.
・More generally every Z-group is metacyclic. A Z-group is a group whose Sylow subgroups are cyclic.

http://mathworld.wolfram.com/MetacyclicGroup.html
Metacyclic Group Wolfram MathWorld

http://rio2016.5ch.net/test/read.cgi/math/1568026331/961

976: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/10/18(金) 15:53:29.16 ID:X/c9sPkS

>>971 追加

ガロアと名の付く数学用語一覧
（これだけで全部じゃないと思うが）（＾＾
なお、ガロアと名はつかないが、ガロアの後、抽象的な群論が活発に研究された
なので、古典ガロア理論を学べば、必然群論も体論も、おそらくは環や、その他もろもろの代数系の学習の助けになるだろう（＾＾；
https://en.wikipedia.org/wiki/Glossary_of_field_theory
Glossary of field theory
(抜粋)
Types of fields
Finite field
A field with finitely many elements. Aka Galois field.

Frobenius field
A pseudo algebraically closed field whose absolute Galois group has the embedding property.[8]

Field extensions
Galois extension
A normal, separable field extension.

Galois theory
Galois extension
A normal, separable field extension.

Galois group
The automorphism group of a Galois extension. When it is a finite extension, this is a finite group of order equal to the degree of the extension. Galois groups for infinite extensions are profinite groups.

Kummer theory
The Galois theory of taking n-th roots, given enough roots of unity. It includes the general theory of quadratic extensions.

Normal basis
A basis in the vector space sense of L over K, on which the Galois group of L over K acts transitively.

Extensions of Galois theory
Inverse problem of Galois theory
Given a group G, find an extension of the rational number or other field with G as Galois group.

Differential Galois theory
The subject in which symmetry groups of differential equations are studied along the lines traditional in Galois theory. This is actually an old idea, and one of the motivations when Sophus Lie founded the theory of Lie groups. It has not, probably, reached definitive form.

Grothendieck's Galois theory
A very abstract approach from algebraic geometry, introduced to study the analogue of the fundamental group.

http://rio2016.5ch.net/test/read.cgi/math/1568026331/976

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