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)
上下前次1-新
通常表示 512ﾊﾞｲﾄ分割ﾚｽ栞
抽出解除ﾚｽ栞

このｽﾚｯﾄﾞは過去ﾛｸﾞ倉庫に格納されています｡
次ｽﾚ検索歴削→次ｽﾚ栞削→次ｽﾚ過去ﾛｸﾞﾒﾆｭｰ

ﾘﾛｰﾄﾞ規制です｡10分ほどで解除するので､他のﾌﾞﾗｳｻﾞへ避難してください｡

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

268: 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) 12:34:53.72 ID:EUeYkluT

>>266

つづき

An example
The following example illustrates the notions introduced above. In order to compute the ramification index of Q(x), where

f(x) = x^3 - x - 1 = 0,
at 23, it suffices to consider the field extension Q23(x) / Q23. Up to 529 = 232 (i.e., modulo 529) f can be factored as

f(x) = (x + 181)(x^2 - 181x - 38) = gh.
Substituting x = y + 10 in the first factor g modulo 529 yields y + 191, so the valuation |?y?|g for y given by g is |?-191?|23 = 1. On the other hand, the same substitution in h yields y2 - 161y - 161 modulo 529. Since 161 = 7?×?23,

|y|h = √?161?23 = 1 / √23.
Since possible values for the absolute value of the place defined by the factor h are not confined to integer powers of 23, but instead are integer powers of the square root of 23, the ramification index of the field extension at 23 is two.

The valuations of any element of F can be computed in this way using resultants. If, for example y = x^2 - x - 1, using the resultant to eliminate x between this relationship and f = x^3 - x - 1 = 0 gives y^3 - 5y^2 + 4y - 1 = 0.
If instead we eliminate with respect to the factors g and h of f, we obtain the corresponding factors for the polynomial for y, and then the 23-adic valuation applied to the constant (norm) term allows us to compute the valuations of y for g and h (which are both 1 in this instance.)

つづく

http://rio2016.5ch.net/test/read.cgi/math/1571400076/268

上下前次1-新書関写板覧索設栞歴

ｽﾚ情報赤ﾚｽ抽出画像ﾚｽ抽出歴の未読ｽﾚ AAｻﾑﾈｲﾙ

ぬこの手ぬこTOP 0.029s