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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266: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/10/27(日) 12:32:42.26 ID:EUeYkluT

つづき

分岐（Ramification）の話
https://en.wikipedia.org/wiki/Algebraic_number_field#Galois_groups_and_Galois_cohomology
Algebraic number field
(抜粋)
In mathematics, an algebraic number field (or simply number field) F is a finite degree (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector space over Q.

The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.

Ramification
https://upload.wikimedia.org/wikipedia/commons/thumb/4/44/Schematic_depiction_of_ramification.svg/300px-Schematic_depiction_of_ramification.svg.png
Schematic depiction of ramification: the fibers of almost all points in Y below consist of three points, except for two points in Y marked with dots, where the fibers consist of one and two points (marked in black), respectively. The map f is said to be ramified in these points of Y.

Ramification, generally speaking, describes a geometric phenomenon that can occur with finite-to-one maps (that is, maps f: X → Y such that the preimages of all points y in Y consist only of finitely many points): the cardinality of the fibers f-1(y) will generally have the same number of points, but it occurs that, in special points y, this number drops. For example, the map

C → C, z → zn
has n points in each fiber over t, namely the n (complex) roots of t, except in t = 0, where the fiber consists of only one element, z = 0.
One says that the map is "ramified" in zero. This is an example of a branched covering of Riemann surfaces.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1571400076/266

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