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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335: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/11/01(金) 07:57:00.74 ID:rcKeDs9u

>>334 追加
https://fr.wikipedia.org/wiki/Corps_fini
Corps fini
(抜粋)
4 Histoire
4.1 Congruences et imaginaires de Galois
4.3 Applications theoriques
（google 英訳）
History
The theory of finite fields first develops, like the study of congruences, on integers and on polynomials, then from the very end of the nineteenth century, as part of a general theory of commutative bodies.

Congruences and imaginations of Galois
The study of the first finite fields is systematically treated, in the form of congruences, by Gauss in his Disquisitiones arithmeticae published in 1801,
but many of these properties had already been established by Fermat, Euler, Lagrange and Legendre, among others.

In 1830 Evariste Galois published28 what is considered as the founding article of the general theory of finite bodies. Galois, who claims to be inspired by Gauss's work on entire congruences, deals with polynomial congruences, for an irreducible polynomial with coefficients taken themselves modulo a prime number p.
More precisely, Galois introduces an imaginary root of a congruence P (x) = 0 modulo a prime number p, where P is an irreducible polynomial modulo p. He notes i this root and works on expressions:
a + a1 i + a2 i2 + ... + an-1 in-1 where n is the degree of P.

Retraduced in modern terms, Galois shows that these expressions form a cardinality body pn, and that the multiplicative group is cyclic (Kleiner 1999,).
He also notes that an irreducible polynomial that has a root in this body, has all its roots in it, that is, it is a normal extension of its first subfield ( Lidl and Niederreiter 1997).
He uses the identity given by what has been called since the Frobenius automorphism (Van der Waerden 1985).
In 1846, Liouville, at the same time as he published Galois' famous memoir on the resolution of polynomial equations, republished this article in his Journal of Pure and Applied Mathematics.
(引用終り)

http://rio2016.5ch.net/test/read.cgi/math/1571400076/335

340: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/11/01(金) 12:28:10.24 ID:jBvN9kSg

>>335

独語 有限体
https://de.wikipedia.org/wiki/Endlicher_K%C3%B6rper
Endlicher Korper
(抜粋)
Inhaltsverzeichnis
1 Beispiel: Der Korper mit 2 Elementen
2 Klassifikation endlicher Korper
3 Multiplikative Gruppe und diskreter Logarithmus
4 Weitere Beispiele
4.1 Der Korper mit 4 Elementen
4.2 Der Korper mit 49 Elementen
4.3 Der Korper mit 25 Elementen
5 Zur historischen Entwicklung

Zur historischen Entwicklung
Dass man mit Zahlen modulo einer Primzahl ?wie mit rationalen Zahlen“ rechnen kann, hatte bereits Gaus gezeigt.[1]
Galois fuhrte in die Rechnung modulo p imaginare Zahlgrosen ein, ganz so wie die imaginare Einheit {i} in den komplexen Zahlen.
Damit hat er wohl als erster Korpererweiterungen von {F}_{p} betrachtet ? wenn auch der abstrakte Korperbegriff erst 1895 durch Heinrich Weber eingefuhrt wurde und Frobenius als Erster diesen 1896 auf endliche Strukturen ausdehnte.
Daneben bzw. zuvor hat offenbar Eliakim Hastings Moore 1893 bereits endliche Korper studiert und den Namen Galois field eingefuhrt.[2]

（google 英訳）
On the historical development
Gauss had already shown that one can count on numbers modulo a prime "as with rational numbers". [1] Galois introduced into the calculation modulo p imaginary numbers, much like the imaginary unit {i} in the complex numbers.
He was probably the first body extension of {F}_{p} - although the abstract concept of the body was first introduced by Heinrich Weber in 1895 and Frobenius was the first to introduce it in 1896 extended to finite structures.
In addition, or before apparently Eliakim Hastings Moore 1893 already studied finite body and introduced the name Galois field.
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1571400076/340

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