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A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

The 1874 Controversy between Jordan and Kronecker

Frédéric Brechenmacher

Université d’Artois Laboratoire de mathématiques de Lens & École polytechnique. Département humanités et sciences sociales

April 6, 2013

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

Introduction

Kronecker 1874b : 1181 (trans. from German) In Mr Jordan’s memoir..., the solution of the first problem is not truly new, the solution of the second problem is missed, and that of the third one is not sufficiently grounded. We should add that actually the third problem includes the two others as particular cases, and that its complete solution stems from Mr Weierstrass’ work of 1868, and can also be derived from my additional contribution to this work. Unless I am very much mistaken, there are serious grounds for questioning M. Jordan’s priority in the invention of his results, should they even be correct...

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

Introduction

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

A crucial episode in the evolution of Jordan’s career

X 1855

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

A crucial episode in the evolution of Jordan’s career

X 1855 Thesis 1860

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

A crucial episode in the evolution of Jordan’s career

X 1855 Thesis 1860 Traité des substitutions et des équations algébriques, 1870

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

Professeur d’analyse, École polytechnique, 1876

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

Professeur d’analyse, École polytechnique, 1876 Academy of Paris, 1881

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

Professeur d’analyse, École polytechnique, 1876 Academy of Paris, 1881 Collège de France, 1883

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

Professeur d’analyse, École polytechnique, 1876 Academy of Paris, 1881 Collège de France, 1883 Director of the Journal de mathématiques pures et appliquées, 1885-1922

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

Jordan : an "algebraic" theory on the model of group theory

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

Jordan : an "algebraic" theory on the model of group theory

Jordan 1873 : 7-11, trans. from French It is known that any bilinear polynomial P = ΣAαβxαyβ (α = 1, 2, . . . , n, β = 1, 2, . . . , n) can be reduced to its canonical form x1y1 + ... + xmym, by linear transformations applied to the two sets of variables x1, ..., xn, y1, ..., yn. We now consider the following questions among the many of this kind :

• 1. To reduce a bilinear polynomial P to its canonical form by applying
• rthogonal substitutions on the two systems of variables x1, ...,xn ;

y1, ..., yn.

• 2. To reduce a bilinear polynomial P to its canonical form by using some

substitutions simultaneously on the x’s and the y’s.

• 3. To reduce simultaneously two bilinear polynomials P and Q to a

canonical form by using some linear substitutions on each set of variables individually.

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

The theory of bilinear forms in Berlin in the 1860s

1866, Christoffel and Kronecker : the foundation of a "theory" devoted to the necessary and sufficient conditions under which P = ΣAαβxαyβ can be transformed into P′ = ΣBαβxαyβ by using linear substitutions.

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

The theory of bilinear forms in Berlin in the 1860s

1866, Christoffel and Kronecker : the foundation of a "theory" devoted to the necessary and sufficient conditions under which P = ΣAαβxαyβ can be transformed into P′ = ΣBαβxαyβ by using linear substitutions. Methods : computations of invariants.

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

The theory of bilinear forms in Berlin in the 1860s

The problem of the simultaneous transformations of two forms P and Q would shortly become the main question of the theory.

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

The theory of bilinear forms in Berlin in the 1860s

The problem of the simultaneous transformations of two forms P and Q would shortly become the main question of the theory. The determinant of the pair P + sQ is a polynomial invariant but the roots of the characteristic equation det(P + sQ) = 0 provides a complete set of invariants only under the condition that no multiple roots exist.

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

The theory of bilinear forms in Berlin in the 1860s

The problem of the simultaneous transformations of two forms P and Q would shortly become the main question of the theory. The determinant of the pair P + sQ is a polynomial invariant but the roots of the characteristic equation det(P + sQ) = 0 provides a complete set of invariants only under the condition that no multiple roots exist. Weierstrass, 1868 : a complete set of invariants for the case det(P + sQ) = 0: the elementary divisors.

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

Jordan’s landing in a berliner courtyard in 1873

Jordan ,1873 : 7-11 . . . the third [problem has already been dealt with] by M. Weierstrass ; but the solutions given by the eminent Berlin geometers are incomplete, [...] Their analysis is moreover quite difficult to follow – especially that of Mr Weierstrass. On the contrary, the new methods that we propose are extremely simple and hold no exceptions. . . The simultaneous reduction of two functions P and Q is a problem identical to the reduction of a linear substitution to its canonical form.

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

The opposition between two theorems for non singular pairs

• f bilinear forms (P, Q)

Weierstrass, 1868, elemen- tary divisors : a complete set of polynomial invariants computed from the deter- minant det(P + sQ) Jordan, 1870 Traité des substitu- tions et des équations algébriques : canonical form theorem for sub- stitutions of linear groups.

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

Introduction

From the standpoint of linear algebra, Jordan’s canonical form theorem for matrices with coefficients belonging to an algebraically closed field is equivalent to Weierstrass’ elementary divisors theorem...

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

Much a do about nothing ?

Much a do about nothing ? (Dieudonné, 1961 ; Hawkins, 1977) Two equivalent theorems of "linear algebra."

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

Much a do about nothing ?

Much a do about nothing ? (Dieudonné, 1961 ; Hawkins, 1977) Two equivalent theorems of "linear algebra." A quarrel of priority ?

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

The quarrel : a chronology

• Jordan. December 1873. "Sur les Polynômes bilinéaires". Note

l’Académie des Sciences de Paris.

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

The quarrel : a chronology

• Jordan. December 1873. "Sur les Polynômes bilinéaires". Note

l’Académie des Sciences de Paris.

• Kronecker. December 22, 1873. "Ueber schaaren von

quadratischen und bilinearen formen". Academy of Berlin.

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

The quarrel : a chronology

Jordan to Kronecker, December 1873 Je ne voudrais pas . . . , que je désire la guerre, et que je préférerais une polémique / guerre/ des dé- bats publics à des explications amicales. Ce n’est pas moi qui ait ouvert les hostilités commencé la polémique. J’ai publié il est vrai (c’était mon droit évident) sans vous consulter des recherches qui complétaient les vôtres [. . . ] Si au lieu de jeter brusquement ce débat dans le public, vous vous étiez adressé à moi [...] j’aurais constaté immédiatement, ce que j’ai reconnu trop tard, que votre méthode de 1868 relu plus attentive- ment votre mémoire de 1868 et constaté, ce que je n’avais pas remarqué à première vue, que les formes bilinéaires non citées dans votre travail, y sont pourtant implicitement comprises.

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

The quarrel : a chronology

December 1873 to March 1874 : private correspondence. mostly about the issue of priority / the collective dimensions of some texts published in Berlin.

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

The quarrel : a chronology

December 1873 to March 1874 : private correspondence. mostly about the issue of priority / the collective dimensions of some texts published in Berlin. A "levis culpa" (Kronecker’s own work in regard with Hermite’s and Serret’s) :

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

The quarrel goes public again

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

The quarrel goes public again

Kronecker 1874 : 367 Should such general expressions be found, one should in every case be able to justify calling all of them canonical forms on the basis of their generality and simplicity ; but if one does not want to stick to the purely formal viewpoint which is often put the fore in the more recent Algebra – certainly not for the greatest advantage of the true knowledge -, one shall not omit to derive the correction of these canonical forms on the basis of inner

• grounds. Truly, these so called canonical or normal forms are

determined only by the orientation of the study, but they should not be seen as the aim of the research. . .

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

The quarrel goes public again

• Kronecker. January – March 1874. "Ueber quadratische und bilineare

Formen". Academy of Berlin.

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

The quarrel goes public again

• Kronecker. January – March 1874. "Ueber quadratische und bilineare

Formen". Academy of Berlin.

• Jordan. March 1874 . "Sur les formes bilinéaires". Publication in the

Journal de Liouville of the memoir announced in 1873

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

The quarrel goes public again

• Kronecker. January – March 1874. "Ueber quadratische und bilineare

Formen". Academy of Berlin.

• Jordan. March 1874 . "Sur les formes bilinéaires". Publication in the

Journal de Liouville of the memoir announced in 1873

• Jordan. March, 2 1874. "Sur la réduction des formes bilinéaires".

Academy of Paris.

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

The quarrel goes public again

• Kronecker. January – March 1874. "Ueber quadratische und bilineare

Formen". Academy of Berlin.

• Jordan. March 1874 . "Sur les formes bilinéaires". Publication in the

Journal de Liouville of the memoir announced in 1873

• Jordan. March, 2 1874. "Sur la réduction des formes bilinéaires".

Academy of Paris.

• Kronecker. April 1874. "Sur les faisceaux de formes quadratiques et

bilinéaires". Academy of Paris.

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

The quarrel goes public again

• Kronecker. January – March 1874. "Ueber quadratische und bilineare

Formen". Academy of Berlin.

• Jordan. March 1874 . "Sur les formes bilinéaires". Publication in the

Journal de Liouville of the memoir announced in 1873

• Jordan. March, 2 1874. "Sur la réduction des formes bilinéaires".

Academy of Paris.

• Kronecker. April 1874. "Sur les faisceaux de formes quadratiques et

bilinéaires". Academy of Paris.

• Kronecker. May 1874. "Nachtrag". Sequel to the memoir of March,

Academy of Berlin.

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

The quarrel goes public again

• Kronecker. January – March 1874. "Ueber quadratische und bilineare

Formen". Academy of Berlin.

• Jordan. March 1874 . "Sur les formes bilinéaires". Publication in the

Journal de Liouville of the memoir announced in 1873

• Jordan. March, 2 1874. "Sur la réduction des formes bilinéaires".

Academy of Paris.

• Kronecker. April 1874. "Sur les faisceaux de formes quadratiques et

bilinéaires". Academy of Paris.

• Kronecker. May 1874. "Nachtrag". Sequel to the memoir of March,

Academy of Berlin.

• Jordan. June 1874. "Sur les systèmes de formes quadratiques". Academy
• f Paris,

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

The quarrel goes public again

• Kronecker. January – March 1874. "Ueber quadratische und bilineare

Formen". Academy of Berlin.

• Jordan. March 1874 . "Sur les formes bilinéaires". Publication in the

Journal de Liouville of the memoir announced in 1873

• Jordan. March, 2 1874. "Sur la réduction des formes bilinéaires".

Academy of Paris.

• Kronecker. April 1874. "Sur les faisceaux de formes quadratiques et

bilinéaires". Academy of Paris.

• Kronecker. May 1874. "Nachtrag". Sequel to the memoir of March,

Academy of Berlin.

• Jordan. June 1874. "Sur les systèmes de formes quadratiques". Academy
• f Paris,
• Jordan. July 1874. "Mémoire sur la réduction et la transformation des

systèmes quadratiques", Journal de Liouville

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

The quarrel goes public again

• Kronecker. January – March 1874. "Ueber quadratische und bilineare

Formen". Academy of Berlin.

• Jordan. March 1874 . "Sur les formes bilinéaires". Publication in the

Journal de Liouville of the memoir announced in 1873

• Jordan. March, 2 1874. "Sur la réduction des formes bilinéaires".

Academy of Paris.

• Kronecker. April 1874. "Sur les faisceaux de formes quadratiques et

bilinéaires". Academy of Paris.

• Kronecker. May 1874. "Nachtrag". Sequel to the memoir of March,

Academy of Berlin.

• Jordan. June 1874. "Sur les systèmes de formes quadratiques". Academy
• f Paris,
• Jordan. July 1874. "Mémoire sur la réduction et la transformation des

systèmes quadratiques", Journal de Liouville

• Kronecker. 1874. "Uber die congruenten Transformationen der bilinearen

formen". Academy of Berlin.

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

Much a do about nothing ?

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

Much a do about nothing ?

Two theorems equivalent from the standpoint of linear algebra...

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

Much a do about nothing ?

Two theorems equivalent from the standpoint of linear algebra... ... which did not exist as a discipline before the 1930s

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

Much a do about nothing ?

Two theorems equivalent from the standpoint of linear algebra... ... which did not exist as a discipline before the 1930s A controversy about the very nature of algebra and its role in mathemaics

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

Much a do about nothing ?

Two theorems equivalent from the standpoint of linear algebra... ... which did not exist as a discipline before the 1930s A controversy about the very nature of algebra and its role in mathemaics Jordan’s practice of canonical reduction vs Kronecker’s practice of invariant computation.

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

Much a do about nothing ?

Two theorems equivalent from the standpoint of linear algebra... ... which did not exist as a discipline before the 1930s A controversy about the very nature of algebra and its role in mathemaics Jordan’s practice of canonical reduction vs Kronecker’s practice of invariant computation. The opposition between two mathematical cultures

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

Much a do about nothing ?

Two theorems equivalent from the standpoint of linear algebra... ... which did not exist as a discipline before the 1930s A controversy about the very nature of algebra and its role in mathemaics Jordan’s practice of canonical reduction vs Kronecker’s practice of invariant computation. The opposition between two mathematical cultures What were the collective dimensions of such cultures ? France vs Germany / an echo of the 1870 Prussian war ?

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics

Much a do about nothing ?

Two theorems equivalent from the standpoint of linear algebra... ... which did not exist as a discipline before the 1930s A controversy about the very nature of algebra and its role in mathemaics Jordan’s practice of canonical reduction vs Kronecker’s practice of invariant computation. The opposition between two mathematical cultures What were the collective dimensions of such cultures ? France vs Germany / an echo of the 1870 Prussian war ? A shared algebraic culture at the start of the controversy

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

The reference to a shared algebraic culture

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

The reference to a shared algebraic culture

Antoine Yvon-Villarceau, 1870, "Note sur les conditions des petites

• scillations d’un corps solide de figure quelconque et la théorie des

équations différentielles linéaires"

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

The reference to a shared algebraic culture

Antoine Yvon-Villarceau, 1870, "Note sur les conditions des petites

• scillations d’un corps solide de figure quelconque et la théorie des

équations différentielles linéaires" Problems of small oscillations in Lagrange’s works about a hundred years before .

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

From the small oscillations of swinging strings to the ones of periodic trajectories

Lagrange, 1766 : small oscillations ξi(t) of a string loaded with n bodies

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

From the small oscillations of swinging strings to the ones of periodic trajectories

Lagrange, 1766 : small oscillations ξi(t) of a string loaded with n bodies by neglecting the non linear terms in the power series developments of the equations of dynamics

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

From the small oscillations of swinging strings to the ones of periodic trajectories

Lagrange, 1766 : small oscillations ξi(t) of a string loaded with n bodies by neglecting the non linear terms in the power series developments of the equations of dynamics In the 1770s, Lagrange and Laplace have transfered this mathematization to the investigation of the "secular inequalities in planetary theory" i.e. to the small oscillations of the planets of the solar system on their orbits.

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

A system of n linear equations with constant coefficients : dξi dt =

• j=1,n

Ai,jξj The integration of the system is based on its decomposition into n independent equations dξi

dt = αiξj

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

A system of n linear equations with constant coefficients : dξi dt =

• j=1,n

Ai,jξj The integration of the system is based on its decomposition into n independent equations dξi

dt = αiξj

a mathematization of the mechanical observation that the

• scillations of a swinging string loaded with n bodies can be

decomposed into the independent oscillations of n strings loaded with a single body (Bernouilli)

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

A system of n linear equations with constant coefficients : dξi dt =

• j=1,n

Ai,jξj The integration of the system is based on its decomposition into n independent equations dξi

dt = αiξj

a mathematization of the mechanical observation that the

• scillations of a swinging string loaded with n bodies can be

decomposed into the independent oscillations of n strings loaded with a single body (Bernouilli) Let S be the periodicity of such a proper oscillation (i.e. an eigenvalue of A − SI) :

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

A system of n linear equations with constant coefficients : dξi dt =

• j=1,n

Ai,jξj The integration of the system is based on its decomposition into n independent equations dξi

dt = αiξj

a mathematization of the mechanical observation that the

• scillations of a swinging string loaded with n bodies can be

decomposed into the independent oscillations of n strings loaded with a single body (Bernouilli) Let S be the periodicity of such a proper oscillation (i.e. an eigenvalue of A − SI) :

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

A system of n linear equations with constant coefficients : dξi dt =

• j=1,n

Ai,jξj The integration of the system is based on its decomposition into n independent equations dξi

dt = αiξj

a mathematization of the mechanical observation that the

• scillations of a swinging string loaded with n bodies can be

decomposed into the independent oscillations of n strings loaded with a single body (Bernouilli) Let S be the periodicity of such a proper oscillation (i.e. an eigenvalue of A − SI) :

• A1,1 − S

A1,2 ... A1,n A2,1 A2,2 − S ... A2,n ... ... ... ... An,1 An,2 ... An,n − S

• = 0

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

• A1,1 − S

A1,2 ... A1,n A2,1 A2,2 − S ... A2,n ... ... ... ... An,1 An,2 ... An,n − S

• = 0

An algebraic equation of degree n : "the equation to the secular inequalities in planetary theory" (the secular equation for short) ξi(t) = C1eα1t + C2eα2t + ... + Cneαnt

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

• A1,1 − S

A1,2 ... A1,n A2,1 A2,2 − S ... A2,n ... ... ... ... An,1 An,2 ... An,n − S

• = 0

An algebraic equation of degree n : "the equation to the secular inequalities in planetary theory" (the secular equation for short) to each root αi of this equation : a proper oscillation ξi(t) = eαit ξi(t) = C1eα1t + C2eα2t + ... + Cneαnt

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

• A1,1 − S

A1,2 ... A1,n A2,1 A2,2 − S ... A2,n ... ... ... ... An,1 An,2 ... An,n − S

• = 0

An algebraic equation of degree n : "the equation to the secular inequalities in planetary theory" (the secular equation for short) to each root αi of this equation : a proper oscillation ξi(t) = eαit If the equation has n distinct roots, one thus gets n independent solutions ξi(t) = C1eα1t + C2eα2t + ... + Cneαnt

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

Lagrange’s algebraic procedure for manipulating linear systems

A rational expression of the coordinates (xαj

i ) of the solutions of

symetric linear systems of n equations with constant coefficients. xαj

i

= ∆1i

∆ S−αj

(αj) Involving ∆(S), the (polynomial) characteristic determinant of the system A, i.e. the one that generates the secular equation : det(A − SI) its (polynomial) successive minors ∆1i(S) (developments / first line and ith column)

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

This expression is provided by the non-zero column of the cofactor matrix of A − SI. For example, given A =

• 1

−1 −1 2 1 1 1

• The characteristic equation :

det(A − SI) = ∆(S) = S(3 − S)(1 − S) ∆11(S) = (1 − S)(2 − S) − 1, ∆12(S) = (1 − S), ∆13(S) = 1 e.g. for the eigenvalue s1 = 1, the coordinates of an eigenvector are : xs1

1 = ∆11 ∆ s−1

(1) = 1 2 , xs1

2 = ∆12 ∆ s−1

(1) = 0 , xs1

3 = ∆13 ∆ s−1

(1) = −1 2

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

The problem of multiple roots

xαj

i

= ∆1i

∆ S−αj

(αj)

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

Mechanical stability and algebraic multiplicity

Lagrange’s 1866 criterion of mechanical stability in function of the algebraic nature of the roots of the secular equation : The mechanical system is stable iff the αi are real, negatives and distinct. In this situation a particular solution has the form sin(αit)

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

Mechanical stability and algebraic multiplicity

Lagrange’s 1866 criterion of mechanical stability in function of the algebraic nature of the roots of the secular equation : The mechanical system is stable iff the αi are real, negatives and distinct. In this situation a particular solution has the form sin(αit) in the case of imaginary roots : some exponential oscillations

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

Mechanical stability and algebraic multiplicity

Lagrange’s 1866 criterion of mechanical stability in function of the algebraic nature of the roots of the secular equation : The mechanical system is stable iff the αi are real, negatives and distinct. In this situation a particular solution has the form sin(αit) in the case of imaginary roots : some exponential oscillations in the case of a multiple root : tsin(αit) : "le temps sort du sinus" and generates some non periodic unbounded oscillations (false :Weierstrass 1858, Jordan 1871)

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

Mechanical stability and algebraic multiplicity

Antoine Yvon-Villarceau, 1870, I claim that this condition is not necessary for the oscillations to remain small. . . .

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

Jordan’s response to Villarceau in 1871

dx1 dt = a1x1 + ... + l1xn dx2 dt = a2x1 + ... + l2xn

...

dxn dt = anx1 + ... + lnxn

Ce problème peut se résoudre très simplement par un procédé identique à celui dont nous nous sommes servi, dans notre Traité des substitutions, pour ramener une substitution linéaire quelconque à sa forme canonique. dy1 dt = σy1, dz1 dt = σz1 + y, du1 dt = σu1 + z1, ..., dw1 dt = σw1 + v1 (...) w1 = eσtψ(t), v1 = eσtψ′(t), ..., y1 = eσtψr(t), ψ(t) étant une fonction entière arbitraire du degré r − 1.

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

Jordan 1871 : first response to Villarceau / application of the canonical form theorem to linear differential equations with constant coefficients

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

Jordan 1871 : first response to Villarceau / application of the canonical form theorem to linear differential equations with constant coefficients Jordan 1872 : second response to Villarceau : in the case of mechanics, systems can always be reduced to a diagonal form / pairs of quadratic forms

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

Jordan 1871 : first response to Villarceau / application of the canonical form theorem to linear differential equations with constant coefficients Jordan 1872 : second response to Villarceau : in the case of mechanics, systems can always be reduced to a diagonal form / pairs of quadratic forms A result already stated by Weierstrass in 1858 in discussing Lagrange’s criterion of stability : multiplicity of roots do not interfere with stability Jordan 1873 : the non symetric case of pairs of bilinear forms

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

Jordan 1871 : first response to Villarceau / application of the canonical form theorem to linear differential equations with constant coefficients Jordan 1872 : second response to Villarceau : in the case of mechanics, systems can always be reduced to a diagonal form / pairs of quadratic forms A result already stated by Weierstrass in 1858 in discussing Lagrange’s criterion of stability : multiplicity of roots do not interfere with stability Jordan 1873 : the non symetric case of pairs of bilinear forms A result already stated by Weierstrass in 1868

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

Two ends given to a shared history

The 1874 controversy : opposing two ends given to a shared history, implicitly referring to the works of d’Alembert Lagrange, Laplace, Cauchy etc.

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

Two ends given to a shared history

The 1874 controversy : opposing two ends given to a shared history, implicitly referring to the works of d’Alembert Lagrange, Laplace, Cauchy etc. It was because of this, that some identities between Jordan’s and Weierstrass’ theorems had arisen between 1870 and 1873.

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

Two ends given to a shared history

The 1874 controversy : opposing two ends given to a shared history, implicitly referring to the works of d’Alembert Lagrange, Laplace, Cauchy etc. It was because of this, that some identities between Jordan’s and Weierstrass’ theorems had arisen between 1870 and 1873. It was also in reference to this history that the quarrel developped

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

Kronecker’s views on the history of generality in algebra

Kronecker 1874a : 367 One is indeed used to discovering essentially new difficulties – especially in algebraic questions -, as soon as ... one forces his way through the surface of this so called generality - which excludes any particularity-, one penetrates the true generality - which encompasses all singularities-, one generally finds the real difficulties of the study, but at the same time one finds the wealth

• f new viewpoints and phenomena which lie in its depths.

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

Kronecker’s views on the history of generality in algebra

Kronecker 1874a : 367 This holds in the few algebraic questions which have been tackled completely and to the smallest details, such as the theory of networks of quadratic forms ... As long as one did not dare to dispense with the hypothesis that the determinant has only unequal factors, one can reach only inadequate results in the well known problem ... which has been dealt with so often over the last century

• ; under this hypothesis, the true viewpoints on the investigation

remained completely unacknowledged.

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

Kronecker’s views on the history of generality in algebra

Kronecker 1874a : 367 Weierstrass’ 1858 work dropped this hypothesis ... the general introduction of the notion of elementary divisor ... the brightest light was shed on the new algebraic configurations, and at the same time by this complete treatment of the subject, the most valuable insights were reached on the theory of the higher invariants, as conceived in their true generality.

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

A shared algebraic culture ; the secular equation

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

A shared algebraic culture ; the secular equation

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

The circulation of Lagrange’s algebraic procedure

Cauchy, 1829, Sur l’équation à l’aide de laquelle on détermine les inégalités séculaires des planètes The problem of finding the principal axis of a conic (with real coefficients) f (x1, x2, ..., xn) = A11x2

1+A22x2 2+...+Annx2 n+2A12x1x2+2A13x1x3+...

To transform it into a sum of squares : "the secular equation"

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

The circulation of Lagrange’s algebraic procedure

Cauchy, 1829, Sur l’équation à l’aide de laquelle on détermine les inégalités séculaires des planètes The problem of finding the principal axis of a conic (with real coefficients) f (x1, x2, ..., xn) = A11x2

1+A22x2 2+...+Annx2 n+2A12x1x2+2A13x1x3+...

To transform it into a sum of squares : "the secular equation" f (x1, x2, ..., xn) = ∆n−1X 2

1 + ∆n−2

∆n−1 X 2

2 + ... + ∆

∆1 X 2

n

with coefficients given by the successive principal minors of the polynomial determinant ∆(S)

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

The circulation of a specific problem : multiple roots

In both expressions : xαj

i

= ∆1i

∆ S−αj

(αj) f (x1, x2, ..., xn) = ∆n−1X 2

1 + ∆n−2

∆n−1 X 2

2 + ... + ∆

∆1 X 2

n

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

The circulation of a specific problem : multiple roots

A problem similar to the one of the root of a negative number / the development of complex analysis : Cauchy’s Residue theory

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

The circulation of a specific problem : multiple roots

A problem similar to the one of the root of a negative number / the development of complex analysis : Cauchy’s Residue theory In the 1850s, different algebraic approaches to the problem of the multiplicity of roots :

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

The circulation of a specific problem : multiple roots

A problem similar to the one of the root of a negative number / the development of complex analysis : Cauchy’s Residue theory In the 1850s, different algebraic approaches to the problem of the multiplicity of roots :

Charles Hermite’s algebraic theory of quadratic forms

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

The circulation of a specific problem : multiple roots

A problem similar to the one of the root of a negative number / the development of complex analysis : Cauchy’s Residue theory In the 1850s, different algebraic approaches to the problem of the multiplicity of roots :

Charles Hermite’s algebraic theory of quadratic forms James Joseph Sylvester’s notions of matrices and minors

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

The circulation of a specific problem : multiple roots

A problem similar to the one of the root of a negative number / the development of complex analysis : Cauchy’s Residue theory In the 1850s, different algebraic approaches to the problem of the multiplicity of roots :

Charles Hermite’s algebraic theory of quadratic forms James Joseph Sylvester’s notions of matrices and minors Karl Weierstrass’s elementary divisors theorem

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

The circulation of a specific problem : multiple roots

A problem similar to the one of the root of a negative number / the development of complex analysis : Cauchy’s Residue theory In the 1850s, different algebraic approaches to the problem of the multiplicity of roots :

Charles Hermite’s algebraic theory of quadratic forms James Joseph Sylvester’s notions of matrices and minors Karl Weierstrass’s elementary divisors theorem Camille Jordan’s canonical form in finite groups theory

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

The circulation of a specific problem : multiple roots

A problem similar to the one of the root of a negative number / the development of complex analysis : Cauchy’s Residue theory In the 1850s, different algebraic approaches to the problem of the multiplicity of roots :

Charles Hermite’s algebraic theory of quadratic forms James Joseph Sylvester’s notions of matrices and minors Karl Weierstrass’s elementary divisors theorem Camille Jordan’s canonical form in finite groups theory

Different lines of developments : a strong structuration of the algebraic methods used at the end of the 19th century

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

The circulation of a specific problem : multiple roots

A problem similar to the one of the root of a negative number / the development of complex analysis : Cauchy’s Residue theory In the 1850s, different algebraic approaches to the problem of the multiplicity of roots :

Charles Hermite’s algebraic theory of quadratic forms James Joseph Sylvester’s notions of matrices and minors Karl Weierstrass’s elementary divisors theorem Camille Jordan’s canonical form in finite groups theory

Different lines of developments : a strong structuration of the algebraic methods used at the end of the 19th century The 1874 controversy : two attempts to give new theoretical identities to wat used to be a broadly shared algebraic culture.

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

Hermite’s algebraic theory of forms

In the 1850s, Hermite and Sylvester have looked for a purely algebraic proof of Sturm theorem through the investigation of the specific case of the secular equation (Sinaceur 1991)

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

Hermite’s approach to Sturm theorem

To the secular equation one can associate a quadratic form : f (x1, x2, ..., xn) = A11x2

1+A22x2 2+...+Annx2 n+2A12x1x2+2A13x1x3+...

which can be transformed into a sum of squares: f (x1, x2, ..., xn) = ∆n−1X 2

1 + ∆n−2

∆n−1 X 2

2 + ... + ∆

∆1 X 2

n

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

Hermite’s approach to Sturm theorem

To the secular equation one can associate a quadratic form : f (x1, x2, ..., xn) = A11x2

1+A22x2 2+...+Annx2 n+2A12x1x2+2A13x1x3+...

which can be transformed into a sum of squares: f (x1, x2, ..., xn) = ∆n−1X 2

1 + ∆n−2

∆n−1 X 2

2 + ... + ∆

∆1 X 2

n

The number of positive and negative signs is an invariant of the quadratic form (Sylvester’s inertia law) that actually provides the number of real distinct roots of the secular equation (and more generally an algebraic proof of Sturm theorem).

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

Hermite’s theory of forms

"The arithmetic theory of quadratic forms" : in Gauss’s legacy :

• perations by substitutions with integer coefficients

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

Hermite’s theory of forms

"The arithmetic theory of quadratic forms" : in Gauss’s legacy :

• perations by substitutions with integer coefficients

"The algebraic theory of forms" : operations by substitutions with real coefficients / algebraic because its role in Sturm theorem

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

Hermite’s theory of forms

"The arithmetic theory of quadratic forms" : in Gauss’s legacy :

• perations by substitutions with integer coefficients

"The algebraic theory of forms" : operations by substitutions with real coefficients / algebraic because its role in Sturm theorem Kronecker 1873, "Sur la theorie algébrique des formes quadratiques"

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Two ends given to a shared history The algebraic theory of forms

Hermite’s theory of forms

"The arithmetic theory of quadratic forms" : in Gauss’s legacy :

• perations by substitutions with integer coefficients

"The algebraic theory of forms" : operations by substitutions with real coefficients / algebraic because its role in Sturm theorem Kronecker 1873, "Sur la theorie algébrique des formes quadratiques" Darboux 1874, "Sur la theorie algébrique des formes quadratiques"

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Kronecker : arithmetics Jordan : algebra

Kronecker : the arithmetic theory of forms / invariant factors

Kronecker 1874b : 415 In the arithmetical theory of forms, one must certainly be satisfied by the indication of a procedure for deciding of the question of the equivalence, and this problem was indeed formulated explicitly in this way too (cf. Gauss : Disquitiones arithmeticae, Sectio V.). The procedure itself is here also based on the transformation to reduced forms: but it must not be forgotten that, in the arithmetic theory, these [reduced forms] have a completely different meaning than the one they have in the Algebra. Indeed, there, the invariants... can be directly defined, although not explicitly but only described as the final result of arithmetic operations ; for much the same is true with most concepts of arithmetic, e.g. even the simple notion greatest common divisor.

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Kronecker : arithmetics Jordan : algebra

Jordan and the generality of algebra : simplicity

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Kronecker : arithmetics Jordan : algebra

Jordan : simplicity / an "algebraic process of reduction"

Jordan’s specific approach to group theory and Galois theory since his 1860 thesis :

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Kronecker : arithmetics Jordan : algebra

Jordan : simplicity / an "algebraic process of reduction"

Jordan’s specific approach to group theory and Galois theory since his 1860 thesis : To reduce the problem of the determination of all solvable groups to a chain of groups with simple quotient groups (i.e. Jordan-Hölder theorem) :

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Kronecker : arithmetics Jordan : algebra

Jordan : simplicity / an "algebraic process of reduction"

Jordan’s specific approach to group theory and Galois theory since his 1860 thesis : To reduce the problem of the determination of all solvable groups to a chain of groups with simple quotient groups (i.e. Jordan-Hölder theorem) : General transitive groups...

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Kronecker : arithmetics Jordan : algebra

Jordan : simplicity / an "algebraic process of reduction"

Jordan’s specific approach to group theory and Galois theory since his 1860 thesis : To reduce the problem of the determination of all solvable groups to a chain of groups with simple quotient groups (i.e. Jordan-Hölder theorem) : General transitive groups... primitive groups

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Kronecker : arithmetics Jordan : algebra

Jordan : simplicity / an "algebraic process of reduction"

Jordan’s specific approach to group theory and Galois theory since his 1860 thesis : To reduce the problem of the determination of all solvable groups to a chain of groups with simple quotient groups (i.e. Jordan-Hölder theorem) : General transitive groups... primitive groups linear groups J’étudie à part ces dernières substitutions, que je représente par la notation suivante :

• x

ax + bx′ + cx′′ + ... x′ a′x + b′x′ + c′x′′ + ... x′′ a′′x + b′′x′ + c′′x′′ + ... .. .....................

• Frédéric Brechenmacher

The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Kronecker : arithmetics Jordan : algebra

Pursuing further the chain of reductions

Théorème de réduction canonique Cette forme simple

• y0, z0, u0, ..., y′

0, ...

K0y0, K0(z0 + y0), ..., K0y′ y1, z1, u1, ..., y′

1, ...

K1y1, K1(z1 + y1), ..., K1y′

1

.... ... v0, ... K ′

0v0, ...

... ...

• à laquelle on peut ramener la substitution A par un choix d’indice

convenable, sera pour nous sa forme canonique.

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Kronecker : arithmetics Jordan : algebra

Conclusions

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Kronecker : arithmetics Jordan : algebra

Conclusions

The 1874 controversy : two mathematical cultures ...

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Kronecker : arithmetics Jordan : algebra

Conclusions

The 1874 controversy : two mathematical cultures ... complete

• ppositions about what should be a "general" treatment :

Kronecker’s effectivity vs Jordan’s reductions to the simplest

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Kronecker : arithmetics Jordan : algebra

Conclusions

The 1874 controversy : two mathematical cultures ... complete

• ppositions about what should be a "general" treatment :

Kronecker’s effectivity vs Jordan’s reductions to the simplest These two mathematical cultures do not correspond to any obvious collective dimension (nations, research school...)

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Kronecker : arithmetics Jordan : algebra

Conclusions

The 1874 controversy : two mathematical cultures ... complete

• ppositions about what should be a "general" treatment :

Kronecker’s effectivity vs Jordan’s reductions to the simplest These two mathematical cultures do not correspond to any obvious collective dimension (nations, research school...) This highlights the complexity of the collective dimensions in which individual creative works take place

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Kronecker : arithmetics Jordan : algebra

Conclusions

The 1874 controversy : two mathematical cultures ... complete

• ppositions about what should be a "general" treatment :

Kronecker’s effectivity vs Jordan’s reductions to the simplest These two mathematical cultures do not correspond to any obvious collective dimension (nations, research school...) This highlights the complexity of the collective dimensions in which individual creative works take place : both the structuring effect of long run developments (the secular equations) as well as of more local ones (Hermite-Kronecker algebraic theory of forms)

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker

A quarrel of priority ? A shared algebraic culture Algebra vs arithmetics Kronecker : arithmetics Jordan : algebra

Conclusions

The 1874 controversy : two mathematical cultures ... complete

• ppositions about what should be a "general" treatment :

Kronecker’s effectivity vs Jordan’s reductions to the simplest These two mathematical cultures do not correspond to any obvious collective dimension (nations, research school...) This highlights the complexity of the collective dimensions in which individual creative works take place : both the structuring effect of long run developments (the secular equations) as well as of more local ones (Hermite-Kronecker algebraic theory of forms) which do not correspond to modern organization of knowledge but can be identified by investigating intertextual relations, i.e. networks of texts.

Frédéric Brechenmacher The 1874 Controversy between Jordan and Kronecker