Réflexions Sur La Résolution Algébrique Des équations

This is a list of publications in mathematics, organized by field.

Some reasons a particular publication might be regarded as important:
*Topic creator – A publication that created a new topic
*Breakthrough – A publication that changed scientific knowledge significantly
*Influence – A publication which has significantly influenced the world or has had a massive impact on the teaching of mathematics.

Among published compilations of important publications in mathematics are ''Landmark writings in Western mathematics 1640–1940'' by Ivor Grattan-Guinness and ''A Source Book in Mathematics'' by David Eugene Smith.

Algebra

Theory of equations

''

Baudhayana Sulba Sutra''

* Baudhayana (8th century BCE)
With linguistic evidence suggesting this text to have been written between the 8th and 5th centuries century BCE, this is one of the oldest mathematical texts. It laid the foundations of Indian mathematics and was influential in South Asia. It was primarily a geometrical text and also contained some important developments, including the list of Pythagorean triples, geometric solutions of linear and quadratic equations and square root of 2.

''

The Nine Chapters on the Mathematical Art''

* ''The Nine Chapters on the Mathematical Art'' (10th–2nd century BCE)
Contains the earliest description of Gaussian elimination for solving system of linear equations, it also contains method for finding square root and cubic root.

''

Arithmetica''

*Diophantus (3rd century CE)
Contains the collection of 130 algebraic problems giving numerical solutions of determinate equations (those with a unique solution) and indeterminate equations.

'' Haidao Suanjing''

*Liu Hui (220-280 CE)

Contains the application of right angle triangles for survey of depth or height of distant objects.

'' Sunzi Suanjing''

*Sunzi (5th century CE)

Contains the earliest description of Chinese remainder theorem.

''

Aryabhatiya''

*Aryabhata (499 CE)

The text contains 33 verses covering mensuration (kṣetra vyāvahāra), arithmetic and geometric progressions, gnomon / shadows (shanku-chhAyA), simple, quadratic, simultaneous, and indeterminate equations. It also gave the modern standard algorithm for solving first-order diophantine equations.

'' Jigu Suanjing''

Jigu Suanjing (626 CE)

This book by Tang dynasty mathematician Wang Xiaotong contains the world's earliest third order equation.

'' Brāhmasphuṭasiddhānta''

*Brahmagupta (628 CE)

Contained rules for manipulating both negative and positive numbers, rules for dealing with the number zero, a method for computing square roots, and general methods of solving linear and some quadratic equations, solution to Pell's equation.

'' Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa'l-muqābala''

* Muhammad ibn Mūsā al-Khwārizmī (820 CE)

The first book on the systematic algebraic solutions of linear and quadratic equations by the Persian scholar Muhammad ibn Mūsā al-Khwārizmī. The book is considered to be the foundation of modern algebra and Islamic mathematics. The word "algebra" itself is derived from the ''al-Jabr'' in the title of the book.

'' Līlāvatī'', '' Siddhānta Shiromani'' and '' Bijaganita''

One of the major treatises on mathematics by Bhāskara II provides the solution for indeterminate equations of 1st and 2nd order.

''

Yigu yanduan''

* Li Ye (13th century)

Contains the earliest invention of 4th order polynomial equation.

''

Mathematical Treatise in Nine Sections''

*Qin Jiushao (1247)

This 13th-century book contains the earliest complete solution of 19th-century Horner's method of solving high order polynomial equations (up to 10th order). It also contains a complete solution of Chinese remainder theorem, which predates Euler and Gauss by several centuries.

'' Ceyuan haijing''

* Li Zhi (1248)

Contains the application of high order polynomial equation in solving complex geometry problems.

'' Jade Mirror of the Four Unknowns''

* Zhu Shijie (1303)

Contains the method of establishing system of high order polynomial equations of up to four unknowns.

'' Ars Magna''

* Gerolamo Cardano (1545)
Otherwise known as ''The Great Art'', provided the first published methods for solving cubic and quartic equations (due to Scipione del Ferro, Niccolò Fontana Tartaglia, and Lodovico Ferrari), and exhibited the first published calculations involving non-real complex numbers.

''Vollständige Anleitung zur Algebra''

* Leonhard Euler (1770)
Also known as Elements of Algebra, Euler's textbook on elementary algebra is one of the first to set out algebra in the modern form we would recognize today. The first volume deals with determinate equations, while the second part deals with Diophantine equations. The last section contains a proof of Fermat's Last Theorem for the case ''n'' = 3, making some valid assumptions regarding $\mathbb(\sqrt)$ that Euler did not prove.

''Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse''

* Carl Friedrich Gauss (1799)
Gauss's doctoral dissertation, which contained a widely accepted (at the time) but incomplete proof of the fundamental theorem of algebra.

Abstract algebra

Group theory

=''Réflexions sur la résolution algébrique des équations''

=
*Joseph Louis Lagrange of a polynomial equation are tied to permutations of the roots of the original equation, laying a more general foundation for what had previously been an ad hoc analysis and helping motivate the later development of the theory of permutation groups, group theory, and Galois theory. The Lagrange resolvent also introduced the discrete Fourier transform of order 3.

''Articles Publiés par Galois dans les Annales de Mathématiques''

* Journal de Mathematiques pures et Appliquées, II (1846)
Posthumous publication of the mathematical manuscripts of Évariste Galois by Joseph Liouville. Included are Galois' papers ''Mémoire sur les conditions de résolubilité des équations par radicaux'' and ''Des équations primitives qui sont solubles par radicaux''.

''Traité des substitutions et des équations algébriques''

*Camille Jordan (1870)
Online version:''
Online version

Traité des substitutions et des équations algébriques (Treatise on Substitutions and Algebraic Equations). The first book on group theory, giving a then-comprehensive study of permutation groups and Galois theory. In this book, Jordan introduced the notion of a simple group and epimorphism (which he called ), proved part of the Jordan–Hölder theorem, and discussed matrix groups over finite fields as well as the Jordan normal form.

''Theorie der Transformationsgruppen''

*Sophus Lie, Friedrich Engel (1888–1893).

Publication data: 3 volumes, B.G. Teubner, Verlagsgesellschaft, mbH, Leipzig, 1888–1893
Volume 1Volume 2Volume 3

The first comprehensive work on transformation groups, serving as the foundation for the modern theory of Lie groups.

''Solvability of groups of odd order''

*Walter Feit and John Thompson (1960)
Description: Gave a complete proof of the solvability of finite groups of odd order, establishing the long-standing Burnside conjecture that all finite non-abelian simple groups are of even order. Many of the original techniques used in this paper were used in the eventual classification of finite simple groups.

Homological algebra

''Homological Algebra''

* Henri Cartan and Samuel Eilenberg (1956)
Provided the first fully worked out treatment of abstract homological algebra, unifying previously disparate presentations of homology and cohomology for associative algebras, Lie algebras, and groups into a single theory.

" Sur Quelques Points d'Algèbre Homologique"

* Alexander Grothendieck (1957)
Often referred to as the "Tôhoku paper", it revolutionized homological algebra by introducing abelian categories and providing a general framework for Cartan and Eilenberg's notion of derived functors.

Algebraic geometry

''Theorie der Abelschen Functionen''

* Bernhard Riemann (1857)
Publication data: ''Journal für die Reine und Angewandte Mathematik''

Developed the concept of Riemann surfaces and their topological properties beyond Riemann's 1851 thesis work, proved an index theorem for the genus (the original formulation of the Riemann–Hurwitz formula), proved the Riemann inequality for the dimension of the space of meromorphic functions with prescribed poles (the original formulation of the Riemann–Roch theorem), discussed birational transformations of a given curve and the dimension of the corresponding moduli space of inequivalent curves of a given genus, and solved more general inversion problems than those investigated by Abel and Jacobi. André Weil once wrote that this paper "''is one of the greatest pieces of mathematics that has ever been written; there is not a single word in it that is not of consequence.''"

''Faisceaux Algébriques Cohérents''

* Jean-Pierre Serre
Publication data: ''Annals of Mathematics'', 1955

''FAC'', as it is usually called, was foundational for the use of sheaves in algebraic geometry, extending beyond the case of complex manifolds. Serre introduced Čech cohomology of sheaves in this paper, and, despite some technical deficiencies, revolutionized formulations of algebraic geometry. For example, the long exact sequence in sheaf cohomology allows one to show that some surjective maps of sheaves induce surjective maps on sections; specifically, these are the maps whose kernel (as a sheaf) has a vanishing first cohomology group. The dimension of a vector space of sections of a coherent sheaf is finite, in projective geometry, and such dimensions include many discrete invariants of varieties, for example Hodge numbers. While Grothendieck's derived functor cohomology has replaced Čech cohomology for technical reasons, actual calculations, such as of the cohomology of projective space, are usually carried out by Čech techniques, and for this reason Serre's paper remains important.

'' Géométrie Algébrique et Géométrie Analytique''

* Jean-Pierre Serre (1956)
In mathematics, algebraic geometry and analytic geometry are closely related subjects, where ''analytic geometry'' is the theory of complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. A (mathematical) theory of the relationship between the two was put in place during the early part of the 1950s, as part of the business of laying the foundations of algebraic geometry to include, for example, techniques from Hodge theory. (''NB'': While analytic geometry as use of Cartesian coordinates is also in a sense included in the scope of algebraic geometry, that is not the topic being discussed in this article.) The major paper consolidating the theory was ''Géometrie Algébrique et Géométrie Analytique'' by Serre, now usually referred to as ''GAGA''. A ''GAGA-style result'' would now mean any theorem of comparison, allowing passage between a category of objects from algebraic geometry, and their morphisms, and a well-defined subcategory of analytic geometry objects and holomorphic mappings.

''Le théorème de Riemann–Roch, d'après A. Grothendieck''

* Armand Borel, Jean-Pierre Serre (1958)
Borel and Serre's exposition of Grothendieck's version of the Riemann–Roch theorem, published after Grothendieck made it clear that he was not interested in writing up his own result. Grothendieck reinterpreted both sides of the formula that Hirzebruch proved in 1953 in the framework of morphisms between varieties, resulting in a sweeping generalization. In his proof, Grothendieck broke new ground with his concept of Grothendieck groups, which led to the development of K-theory.

''

Éléments de géométrie algébrique''

* Alexander Grothendieck (1960–1967)
Written with the assistance of Jean Dieudonné, this is Grothendieck's exposition of his reworking of the foundations of algebraic geometry. It has become the most important foundational work in modern algebraic geometry. The approach expounded in EGA, as these books are known, transformed the field and led to monumental advances.

'' Séminaire de géométrie algébrique''

* Alexander Grothendieck et al.
These seminar notes on Grothendieck's reworking of the foundations of algebraic geometry report on work done at IHÉS starting in the 1960s. SGA 1 dates from the seminars of 1960–1961, and the last in the series, SGA 7, dates from 1967 to 1969. In contrast to EGA, which is intended to set foundations, SGA describes ongoing research as it unfolded in Grothendieck's seminar; as a result, it is quite difficult to read, since many of the more elementary and foundational results were relegated to EGA. One of the major results building on the results in SGA is Pierre Deligne's proof of the last of the open Weil conjectures in the early 1970s. Other authors who worked on one or several volumes of SGA include Michel Raynaud, Michael Artin, Jean-Pierre Serre, Jean-Louis Verdier, Pierre Deligne, and Nicholas Katz.

Number theory

'' Brāhmasphuṭasiddhānta''

*Brahmagupta (628)
Brahmagupta's Brāhmasphuṭasiddhānta is the first book that mentions zero as a number, hence Brahmagupta is considered the first to formulate the concept of zero. The current system of the four fundamental operations (addition, subtraction, multiplication and division) based on the Hindu-Arabic number system also first appeared in Brahmasphutasiddhanta. It was also one of the first texts to provide concrete ideas on positive and negative numbers.

''De fractionibus continuis dissertatio''

*Leonhard Euler (1744)
First presented in 1737, this paper provided the first then-comprehensive account of the properties of continued fractions. It also contains the first proof that the number e is irrational.

''Recherches d'Arithmétique''

*Joseph Louis Lagranges to handle the general problem of when an integer is representable by the form $ax^2 + by^2 + cxy$. This included a reduction theory for binary quadratic forms, where he proved that every form is equivalent to a certain canonically chosen reduced form.

''

Disquisitiones Arithmeticae''

*Carl Friedrich Gauss (1801)
The ''Disquisitiones Arithmeticae'' is a profound and masterful book on number theory written by German mathematician Carl Friedrich Gauss and first published in 1801 when Gauss was 24. In this book Gauss brings together results in number theory obtained by mathematicians such as Fermat, Euler, Lagrange

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