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676: １３２人目の素数さん [] 2023/02/21(火) 07:51:54.64 ID:DYKCwkFh

Lagrange resolvent
原書 仏語かな？

Reflexions sur la resolution algebrique des equations, 1771. Lagrange
https://fr.wikipedia.org/wiki/Joseph-Louis_Lagrange
Principales publications
Reflexions sur la resolution algebrique des equations, 1771.
Ce memoire a inspire Abel et Galois.

http://sites.mathdoc.fr/cgi-bin/oeitem?id=OE_LAGRANGE__3_205_0
Gallica-Math: ?uvres completes
Joseph Louis de Lagrange
Reflexions sur la resolution algebrique des equations
Document (Gallica)
?uvres completes, tome 3, 205-421 (volume)
Nouveaux memoires de l'Academie royale des sciences et belles-lettres de Berlin, annees 1770 et 1771
・Section premiere. De la resolution des equations du troisieme degre 207-254 | Document
・Section seconde. De la resolution des equations du quatrieme degre 254-304 | Document
・Section troisieme. De la resolution des equations du cinquieme degre et des degres ulterieurs 305-355 | Document
・Section quatrieme. Conclusion des reflexions precedentes, avec quelques remarques generales sur la transformation des equations, et sur leur reduction ou abaissement a un moindre degre 355-421 | Document
http://gallica.bnf.fr/ark:/12148/bpt6k229222d/f208

http://rio2016.5ch.net/test/read.cgi/math/1615510393/676

678: １３２人目の素数さん [] 2023/02/21(火) 12:02:02.32 ID:8nIQkhq9

>>676

追加

https://en.wikipedia.org/wiki/Lagrange%27s_theorem_(group_theory)
Lagrange's theorem (group theory)

History
Lagrange himself did not prove the theorem in its general form. He stated, in his article Reflexions sur la resolution algebrique des equations,[3] that if a polynomial in n variables has its variables permuted in all n! ways, the number of different polynomials that are obtained is always a factor of n!. (For example, if the variables x, y, and z are permuted in all 6 possible ways in the polynomial x + y - z then we get a total of 3 different polynomials: x + y - z, x + z - y, and y + z - x. Note that 3 is a factor of 6.) The number of such polynomials is the index in the symmetric group Sn of the subgroup H of permutations that preserve the polynomial. (For the example of x + y - z, the subgroup H in S3 contains the identity and the transposition (x y).) So the size of H divides n!. With the later development of abstract groups, this result of Lagrange on polynomials was recognized to extend to the general theorem about finite groups which now bears his name.

In his Disquisitiones Arithmeticae in 1801, Carl Friedrich Gauss proved Lagrange's theorem for the special case of
(Z/pZ)^*, the multiplicative group of nonzero integers modulo p, where p is a prime.[4] In 1844, Augustin-Louis Cauchy proved Lagrange's theorem for the symmetric group Sn.[5]

Camille Jordan finally proved Lagrange's theorem for the case of any permutation group in 1861.[6]

http://rio2016.5ch.net/test/read.cgi/math/1615510393/678

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