ガロア第一論文及びその関連の資料スレ

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90: １３２人目の素数さん [] 2023/01/30(月) 11:01:53.64 ID:ft46ux2X

>>89
つづき

https://www.ias.ac.in/listing/articles/reso/004/10
The Last Mathematical Testament of Galois Indian Academy of Sciences
Classics Volume 4 Issue 10 October 1999 pp 93-100

https://www.ias.ac.in/article/fulltext/reso/004/10/0093-0100
The Last Mathematical Testament of Galois
Evariste Galois's last mathematical testament in the form ofa letter to his friend Auguste Chevallier is
reproduced here in English translation I.

P3
The last application of the theory of equations is related to the modular equation of elliptic functions.

P5
For p = 7 we find a group of (p + 1) (p - 1) /2 permutations, where
∞ 1 2 4
are respectively related to
0 3 6 5.
This group has its substitutions of the form
略
b being the letter corresponding to c, and a a letter which is a residue or non-residue
according as c.

For p = 11, the same substitutions take place with the same notations,
∞ 1 3 4 5 9
are respectively related to
o 2 6 8 10 7.
Thus, for the case of p = 5,7,11, the modular equation is reduced to degree p.
In all rigor, this reduction is not possible in the higher cases.

The third paper concerns the integrals.
We know that a. sum of terms of the same elliptic function is always reduced to a
single term plus algebraic or logarithmic quantities.

https://www.アマゾン
近世数学史談 (岩波文庫) Paperback Bunko ? August 18, 1995
by 高木 貞治

http://rio2016.5ch.net/test/read.cgi/math/1615510393/90

110: １３２人目の素数さん [] 2023/01/31(火) 15:55:49.87 ID:tkHk7/Du

>>75
>ガロアより以前に置換群論において正規部分群という概念を思いついた
>という人は居ないのだろうか？

ガロアは、Chevallierへの手紙（下記）で
・正規部分群について明記している
（This is called proper decomposition：G = H + H S + H S' + ・・とG = H +TH +T'H +・・とが一致するとき）
・”If each of these groups has a prime number of permutations then the equation will be solvable by radicals; otherwise, not.”と明記している
・The smallest number of permutations that an indecomposable group can have,when this number is not a prime number, is 5・4・3.（＝位数60のA5(交代群)）と明記している

Chevallierへの手紙は、明らかにガロア理論の創始！
（これより以前は、アーベルの方程式論が最前線です）

（参考）
　>>90より
https://www.ias.ac.in/article/fulltext/reso/004/10/0093-0100
The Last Mathematical Testament of Galois
Evariste Galois's last mathematical testament in the form ofa letter to his friend Auguste Chevallier is
reproduced here in English translation I.

Ｐ１
The second contains rather interesting applications from the theory of equations.Here is a summary of the most important ones:
1. According to the propositions II and III of the first paper, one sees a great difference between adjoining, to an equation, . one of the roots or all the roots of an auxiliary equation.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1615510393/110

115: １３２人目の素数さん [sage] 2023/01/31(火) 21:02:38.92 ID:FSzGv1IG

>>93 補足
>>>楕円曲線の等分問題で、
>>p = 11の解法を取り上げている
>>　それ、モジュラー方程式の話
>>　モジュラー方程式、わかってる？
>ありがとう
>笠原乾吉先生
>「モジュラー方程式という語は19世紀数学にはよく登場するが、日本数学会「数学辞典」には見つからないほどに、今日では忘れられている」
>これが、1990年

今頃気づいたが
下記ガロア第一論文でも
”The last application of the theory of equations is related to the modular. equation of elliptic functions.”
と使われているね
”related to the modular. equation of elliptic functions.”だね
レムニスケートの等分と類似ないし同じ意味だね

(参考)（>>90より再録）
https://www.ias.ac.in/article/fulltext/reso/004/10/0093-0100
The Last Mathematical Testament of Galois
Evariste Galois's last mathematical testament in the form ofa letter to his friend Auguste Chevallier is
reproduced here in English translation I.

P3
The last application of the theory of equations is related to the modular. equation of elliptic functions.
We show that the group of the equation which has for roots the sine of the amplitude of p2 - 1 divisions of a period is:

http://rio2016.5ch.net/test/read.cgi/math/1615510393/115

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