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Number Theory or arithmetic, as some prefer to call it, is the oldest, purest, liveliest, most elementary yet sophisticated field of mathematics. It is no coincidence that the fundamental science of numbers has come to be known as the "Queen of Mathematics." Indeed some of the most complex conventions of the mathematical mind have evolved from the study of basic problems of number theory.

André Weil, one of the outstanding contributors to number theory, has written an historical exposition of this subject; his study examines texts that span roughly thirty-six centuries of arithmetical work — from an Old Babylonian tablet, datable to the time of Hammurapi to Legendre’s Essai sur la Théorie des Nombres (1798). Motivated by a desire to present the substance of his field to the educated reader, Weil employs an historical approach in the analysis of problems and evolving methods of number theory and their significance within mathematics. In the course of his study Weil accompanies the reader into the workshops of four major authors of modern number theory (Fermat, Euler, Lagrange and Legendre) and there he conducts a detailed and critical examination of their work. Enriched by a broad coverage of intellectual history, Number Theory represents a major contribution to the understanding of our cultural heritage.

Customer Reviews (2)

Weil's Number Theory
This item is of very great interest to me. It covers thoroughly, in brilliant exegesis, the whole history and mathematics of number theory from Hammurapi to Legendre at a level suitable for the general reader with an interest in number theory. The author was one of the pioneers in this field.

Respectable and enjoyable guide to pre-Gaussian number theory
When someone like Weil sets out to write a history of number theory it is destined to be the standard reference for decades to come. But this is not only an authoritative reference everyone loves to cite--it is also delightfully readable. It is not a substitute for a textbook (although Weil hints at this possibility is the preface), but even for readers with only a modest background in number theory this book will be a source of insight and joy.

Chapter 1 "Protohistory" treats briefly some of the scattered pre-Fermat attempts, which helped form the Diophantine tradition of what would constitute the staple problems of number theory--Pythagorean triples, sums of squares, Pell's equation, such things. These seeds blossomed in the hands of Fermat (chapter 2), with whom we start to see the formation of a coherent theory of numbers with some basic tools: infinite descent, modulo arguments, a precursor of elliptic curve arithmetic, etc. Fermat rarely wrote things down properly, and Euler (chapter 3) had to work hard to prove his theorems and conjectures, in the process adding some ideas of his own (the group theoretic core of modulo arithmetic and Z/pZ, auxiliary functions such as the phi function, etc.). Euler's further investigations along these lines also left many valuable ideas for future mathematicians such as the crystallisation of the importance of quadratic forms (taken up by Lagrange, chaper 4; later perfected by Gauss) and the statement of the law of quadratic reciprocity (taken up by Legendre, chapter 4; later proved in full by Gauss). Also highly decisive for the future development of number theory was Euler's bringing in of analytic ideas into number theory, in particular elliptic integrals (whose deep importance was later revealed by Jacobi) and the zeta function and L-series (whose deep importance was later revealed by Dirichlet and Riemann). ... Read more

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"Extremely readable recollections of the author... A rare testimony of a period of the history of 20th century mathematics. Includes very interesting recollections on the author's participation in the formation of the Bourbaki Group, tells of his meetings and conversations with leading mathematicians, reflects his views on mathematics. The book describes an extraordinary career of an exceptional man and mathematicians. Strongly recommended to specialists as well as to the general public."

EMS Newsletter (1992)

"This excellent book is the English edition of the author's autobiography. … This very enjoyable reading is recommended to all mathematicians."
Acta Scientiarum Mathematicarum (1992)

Customer Reviews (5)

The recollections of a great mathematician
The title is a little misleading since there is not much math in the book, but a lot of personal stories about the intellectual life and travels of A. Weil. Through them we can glimpse his personality and thus the book will be interesting to mathematicians and historians of mathematics. The reader can detect a completely undogmatic mind, skeptical about justice and politics and with a rather ironic sense of humour. His stay in jail for not reporting for duty at the start of WWII was one of his most productive periods.

He recalls, but he is not able to give a concrete date, the day when H. Cartan and he founded Bourbaki. One of the funniest anecdotes is when Cartan receives a call from a Greek whose name is Bourbaki thinking it is a joke, but they become friends and he is even invited to some of their meetings.

A. Weil was not only an outstanding mathematician but a man of a wide culture and a polyglot. He studied sanskrit to read the great books of Indian literature and spent a couple of years in India. Another interesting story is when he is taken by a Russian spy in Finland and he is saved in extremis from execution by a chance meeting of Nevalinna with the chief of police.

He eventually moved to the US, although the first years must have been very frustrating teaching at second rates colleges where when he provided a proof the students would ask: "Is it going to be in the exam?"

The book ends in the fifties when he is appointed a professor at the University of Chicago after a stay in Sao Paulo.

To sum up, a fascinating personality that had a fascinating although not always easy life.

Light reading!
There is only a small number of autobiographies by mathematicians. Andre Weil was a giant in math. His autobio, written late in life, is fun to read. Weil has strong opinions that may perhaps not appeal to all. Even so, the book light reading, agreeing or not; and it fun too. For me it was a page-turner. To others perhaps a little pompous. Judge for yourself. While perhaps self-absorbed, I think Weil in his autobio gives personal and fresh insight into the tumultuous period in history, between the two World Wars in Europe, as it relates to math. The main part of the book covers Weil's life before he came to the US.
Weil had a monumental impact on math, and he also wrote some lovely history of math books, --number theory; and then of course some specialized books, that are corner stones in math, but not especially easy to read, at least for beginners. But Andre Weil is a central figure in math. His younger sister Simone Weil was an author and philosopher, and a political activist on the left in French politics in the 1930ties. She died young.

Light reading!
There is only a small number of autobiographies by mathematicians. Andre Weil was a giant in math. His autobio, written late in life, is fun to read. Weil has strong opinions that may perhaps not appeal to all. Even so, the book light reading, agreeing or not; and it fun too. For me it was a page-turner. To others perhaps a little pompous. Judge for yourself. While perhaps self-absorbed, I think Weil in his autobio gives personal and fresh insight into the tumultuous period in history, between the two World Wars in Europe, as it relates to math. The main part of the book covers Weil's life before he came to the US.
Weil had a monumental impact on math, and he also wrote some lovely history of math books, --number theory;and then of course some specialized books, that are corner stones in math, but not especially easy to read, at least for beginners. But Andre Weil is a central figure in math. His younger sister Simone Weil was an author and philosopher, and a political activist on the left in French politics in the 1930ties. She died young.

Quite some autocensorship
Weil did indeed lead a colorful life, and he certainly was a dominant figure in maths. But if one writes an autobiography, he should be honest when speaking both of his best and his worst. Unlike this, Weil strictlyrestrains from the reader the chapters of his life which he is not so proudof. I will name just two. First, in the book his wife (with a young sonfrom her previous marriage) simply occurs in his life, and the reader isnever told how and where they met, and who was the husband she abandonedbecause of this acquaintance (I here only comment that it was anotherFrench mathematician of the period, and earlier a friend of Weil, but he isnot mentioned anywhere in the story). And second, his evading the militaryservice; surely, most of us do not want to die, but to explain that we donot want to serve because we have a hinduistic view of the World is anoverused trick (though perhaps it was not overused in those times). One sodevoted to that view would definitely not feel at home in Princeton. Inaddition, in his well-known style Weil is extremely critical of theprofessional qualities of practically everyone, with exception of his"family" - the Bourbaki. Most of the textbooks by other peoplewere awful, the knowledge of most other people catastrophicallyincomplete.

Still, there are some interesting facts for those who want toknow what a life of a mathematician can involve, and two stars are justlyearned. Perhaps this would have been a better book has it been written morein the spirit of Hardy's Apology, omitting the family matters and focusingon what the title promises.

Excellent!
Weil led a colorful, fascinating life.He became a well-known mathematician at an early age, and mastered several languages.Additionally, he traveled around the world, and narrowly escaped execution. His autobiography allows the reader to come to know him for himself,rather than for his work.If the reader pays close attention, he/she willsee his subtle humor, which is cleverly woven in to his life story. ... Read more

Customer Reviews (1)

A warning to beginners
Experts find this a very good book, and I rate it on their advice. But others need to understand that Weil is making a bit of a joke with the title. This book is "basic" in the sense that it proves the theorems that Weil feels organize and clarify all of number theory--the "basic" theorems in that sense. It is an introduction to class field theory.

As Weil says at the start of the book, it has few prerequisites in algebra or number theory, except that the reader is presumed familiar with the standard theorems on locally compact Abelian groups, and Pontryagin duality and Haar measures on those groups. This part is not a joke.

If you want to really understand class field theory this may be a good book. (I am reliably told it is.) But Weil deliberately avoids using many ideas that are now standard: geometric ideas such as group schemes, and especially cohomological methods.

Beginners studying algebraic numbers do not need this book. Weil recommends Hecke ALGEBRAIC NUMBERS for such readers, and that is a terrific book. To learn class field theory today you'd probably do better with and Cassels and Frohlich ALGEBRAIC NUMBER THEORY, which Weil also recommends in a note to the second edition of this book. ... Read more

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A. Fröhlich, Bulletin of the London Mathematical Society, 1978: "As a contribution to the history of mathematics, this is a model of its kind. While adhering to the basic outlook of Eisenstein and Kronecker, it provides new insight into their work in the light of subsequent developments, right up to the present day. As one would expect from this author, it also contains some pertinent comments looking into the future. It is not however just a chapter in the history of our subject, but a wide-ranging survey of one of the most active branches of mathematics at the present time. The book has its own very individual flavour, reflecting a sort of combined Eisenstein-Kronecker-Weil personality. Based essentially on Eisenstein's approach to elliptic functions via infinite series over lattices in the complex plane, it stretches back to the very beginnings on the one hand and reaches forward to some of the most recent research work on the other. (...) The persistent reader will be richly rewarded." ... Read more

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This classic is one of the cornerstones of modern algebraic geometry. At the same time, it is entirely self-contained, assuming no knowledge whatsoever of algebraic geometry, and no knowledge of modern algebra beyond the simplest facts about abstract fields and their extensions, and the bare rudiments of the theory of ideals. ... Read more

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Articles by twenty-six leading mathematicians, all reflecting the influence of one of the seminal figures in modern mathematics, Andre Weil.

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From the reviews:

"…All of Weil’s works except for books and lecture notes are compiled here, in strict chronological order for easy reference.

But the value … goes beyond the convenience of easy reference and accessibility. In the first place, these volumes contain several essays, letters, and addresses which were either published in obscure places (…) or not published at all.

Even more valuable are the lengthy commentaries on many of the articles, written by Weil himself. These remarks serve as a guide, helping the reader place the papers in their proper context. Moreover, we have the rare opportunity of seeing a great mathematician in his later life reflecting on the development of his ideas and those of his contemporaries at various stages of his career.

The sheer number of mathematical papers of fundamental significance would earn Weil’s Collected Papers a place in the library of a mathematician with an interest in number theory, algebraic geometry, representations theory, or related areas. The additional import of the mathematical history and culture in these volumes makes them even more essential."

Neal Koblitz in Mathematical Reviews

"…André Weil’s mathematical work has deeply influenced the mathematics of the twentieth century and the monumental (...) "Collected papers" emphasize this influence."

O. Fomenko in Zentralblatt der Mathematik

Also available:

Volume I (1926-1951) ISBN 978-3-540-85888-1

Volume II (1951-1964) ISBN 978-3-540-87735-6

Customer Reviews (1)

Mel Nathanson said
He is a well-known analytic number theorist. In reminiscing about Stanley Tennenbaum at the memorial, Mel told of a number theory course he took from Stanley at U. Rochester:

"The main text was a magical set of lecture notes by Andr'e Weil, 28 pages of typewritten, double-spaced
mimeographed pages. Many years later I was friendly with Morris Schreiber, a professor at Rockefeller
University and also a friend of Stan. I gave Weil's notes to Moe, and he passed them on to Walter Kaufman-
Buhler, then the mathematics editor of Springer in New York. Springer subsequently published them in a
small softcover volume, Number Theory for Beginners, which was supplemented by a set of exercises written
by Maxwell Rosenlicht, who was, evidently, Weil's teaching assistant when he taught the course in Chicago.
Stan described the origin of the notes of Weil as follows: Weil taught an undergraduate algebra (not number
theory) course one summer in Chicago, and these notes were intended to indicate what Weil thought an
undergraduate course in abstract algebra should be. It was in opposition to the then-standard algebra text
by Birkhoff and Maclane, which Weil (according to Stanley) thought was terrible."

... Read more

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This volume contains the original lecture notes presented by A. Weil in which the concept of adeles was first introduced, in conjunction with various aspects of C.L. Siegel’s work on quadratic forms. These notes have been supplemented by an extended bibliography, and by Takashi Ono’s brief survey of subsequent research.

Serving as an introduction to the subject, these notes may also provide stimulation for further research.

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This volume contains the work of the great Swiss mathematician on differential geometry, a field marked by some of his greatest achievements. Between 1690 and 1700, Jacob Bernoulli published twelve treatises in the scientific journal Acta Eruditorum on the use of infinitesimal methods to answer geometrical questions. Preparatory notes for most of these papers and on many other themes are found in Bernoulli's scientific diary Meditationes, from which twentynine texts are published here for the first time. Among the curves considered are the isochrones (lines of constant descent), the parabolic spiral, the loxodrome, the cycloid, the tractrix, and the logarithmic spiral (Bernoulli's spira mirabilis, which also adorns his tombstone). The description of these curves by differential equations and by geometrical constructions, their rectification and quadrature, and the determination of their evolutes and caustics offered Bernoulli and his colleagues a range of challenging problems, many of them relevant for mechanical or optical applications. The French mathematician André Weil, who lived in the United States until his recent death, has greatly influenced 20th century mathematics, among other things, as a founding member of the Bourbaki group. For many years he has pursued intensive studies of the history of mathematics, especially number theory and algebraic geometry. Weil's introduction to this volume places Jacob Bernoulli's contribution to differential geometry in a line of development from Descartes, Huygens and Barrow through Newton's und Leibniz's epochal innovations right up to the codification of the subject by Euler. Martin Mattmüller, secretary of the Bernoulli Edition at Basel, edited the source text. His commentaries consider particular topics in differential geometry with reference to their historical context at the end of the 17th century. ... Read more

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Jacob Bernoulli (1654-1705), founder of Basel's dynasty of mathematicians, left a relatively small body of correspondence for posterity. Published here for the first time are all 50 preserved letters sent to or by Jacob Bernoulli, and documentary evidence is presented of some l00 further letters no longer extant. The correspondence with Leibniz takes up the bulk of the volume and is also the most substantial scientifically. Topics range from Bernoulli's theories of elasticity and probability, to Leibniz' "Dynamik", to the integration of algebraic functions, to controversies over differential calculus and calculus of variations. The exchange offers rare insights into the personal histories and working habits of these two correspondents. Science politics dominates the brief but intensive correspondence (1700-01) with Geneva's Nicolas Fatio de Duillier, a leading exponent of Newtonian mathematics. Their letters broach Leibniz' claim to mathematical leadership, professional disputes between the brothers Jacob and Johann, and Fatio's theory of gravitation. Bernoulli's letters to teachers and friends, and business letters to Otto Mencke, publisher of Acta Eruditorum, shed further light on the until now little-researched life of Basel's famed mathematician. ... Read more

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Chapters: Alexander Grothendieck, André Weil, Nicolas Bourbaki, Séminaire Nicolas Bourbaki, Serge Lang, Jean-Pierre Serre, Claude Chevalley, Laurent Schwartz, Henri Cartan, Jean Dieudonné, Jacques Tits, Samuel Eilenberg, Armand Borel, John Tate, Charles Ehresmann, Szolem Mandelbrojt, Pierre Samuel, René de Possel, Roger Godement, Jacques Dixmier, Pierre Cartier, Jean-Louis Koszul, Jean Delsarte, François Bruhat. Source: Wikipedia. Pages: 119. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: Alexander Grothendieck (born March 28, 1928 in Berlin, Germany) is one of the most influential mathematicians of the twentieth century. He is known principally for his revolutionary advances in algebraic geometry, and also for major contributions to number theory, category theory and homological algebra, and his early achievements in functional analysis. He was awarded the Fields Medal in 1966. He was co-awarded the Crafoord Prize with Pierre Deligne in 1988, but Grothendieck declined this. He is noted for his mastery of abstract approaches to mathematics and his perfectionism in matters of formulation and presentation. In particular, he demonstrated the ability to derive concrete results using only very general methods. Relatively little of his work after 1960 was published by the conventional route of the learned journal, circulating initially in duplicated volumes of seminar notes; his influence was to a considerable extent personal, on French mathematics and the Zariski school at Harvard University. He retired in 1988 and within a few years became reclusive. Grothendieck's early mathematical work was done in functional analysis between 1949 and 1953 working on his doctoral thesis in Nancy, supervised by Jean Dieudonné and Laurent Schwartz. His key contributions include topological tensor products of to...More: http://booksllc.net/?id=2042 ... Read more

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Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Chapters: Alexander Grothendieck, Henri Poincaré, André Weil, Heisuke Hironaka, Vladimir Voevodsky, Elmer Rees, David Mumford, Igor Shafarevich, Jean-Pierre Serre, E. H. Moore, Patrick Du Val, Yozo Matsushima, Pierre Deligne, Oscar Zariski, Kunihiko Kodaira, Francesco Severi, Colin Mclarty, Steven Kleiman, Oleg Viro, S. S. Abhyankar, Shigefumi Mori, William Fulton, Miles Reid, David Eisenbud, Ravi Vakil, Joe Harris, Alfred Clebsch, Vladimir Rokhlin, Pierre Samuel, Frances Kirwan, Michael Artin, Geir Ellingsrud, Erich Kähler, Max Deuring, Nicholas Shepherd-Barron, Suren Arakelov, Jakob Rosanes, Pasquale Del Pezzo, Robin Hartshorne, Lothar Göttsche, Andrei Suslin, Francis Sowerby Macaulay, Victor Flynn, Giusto Bellavitis, Georges Henri Halphen, Rahul Pandharipande, Claire Voisin, Ragni Piene, Andrzej Białynicki-Birula, Geoffrey Horrocks. Source: Wikipedia. Free updates online. Not illustrated. Excerpt: Jules Henri Poincaré (29 April 1854 17 July 1912) (French pronunciation: ) was a French mathematician, theoretical physicist, and a philosopher of science. He is often described as a polymath, and in mathematics as The Last Universalist, since he excelled in all fields of the discipline as it existed during his lifetime. As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics. He was responsible for formulating the Poincaré conjecture, one of the most famous problems in mathematics. In his research on the three-body problem, Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory. He is considered to be one of the founders of the field of topology. Poincaré introduced the modern principle of relativity and was the first to...More: http://booksllc.net/?id=48740 ... Read more

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Les achats comprennent une adhésion à l'essai gratuite au club de livres de l'éditeur, dans lequel vous pouvez choisir parmi plus d'un million d'ouvrages, sans frais. Le livre consiste d'articles Wikipedia sur : Laurent Schwartz, Jean-Pierre Serre, Nicolas Bourbaki, Alexandre Grothendieck, Henri Cartan, André Weil, Éléments de Mathématique, Alain Connes, Szolem Mandelbrojt, Roger Godement, Jean Dieudonné, Séminaire Nicolas Bourbaki, Adrien Douady, Bourbakisme, Charles Ehresmann, Claude Chevalley, Pierre Samuel, Armand Borel, Jean-Christophe Yoccoz, Yves Colin de Verdière, René de Possel, Joseph Oesterlé, Jean Delsarte, Pierre Cartier. Non illustré. Mises à jour gratuites en ligne. Extrait : Nicolas Bourbaki est un mathématicien imaginaire, sous le nom duquel un groupe de mathématiciens francophones, formé en 1935 à Besse-et-Saint-Anastaise (Besse-en-Chandesse à l'époque) en Auvergne sous l'impulsion d'André Weil, a commencé à écrire et éditer des textes mathématiques à la fin des années 1930. L'objectif premier était la rédaction d'un traité d'analyse. Le groupe s'est constitué en association, l'Association des amis de Nicolas Bourbaki, le 30 août 1952. Sa composition a évolué avec un renouvellement constant de générations. Sous le nom Nicolas Bourbaki fut publiée une présentation cohérente des mathématiques, appuyée sur la notion de structure, dans une série d'ouvrages sous le titre Éléments de mathématique. Cette œuvre est à ce jour inachevée. Elle a eu une influence sur l'enseignement des mathématiques et sur l'évolution des mathématiques du siècle. Toutefois, elle connaît de nombreuses critiques : incompatibilité entre le formalisme retenu et la théorie des catégories, , manque d'exemples, incompréhension des étudiants, etc. L'activité du groupe cependant a dépassé la rédaction d'ouvrages, par exemple avec l'organisation des séminaires Bourbaki. Le nom de famille Bourbaki était le nom emprunté par Raoul Husson en 1923 lor...http://booksllc.net/?l=fr ... Read more

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Chapters: André Weil, Khwaja Ahmad Abbas, M. A. G. Osmani, Sheikh Abdullah, Mushirul Hasan, Liaquat Ali Khan, Salimuzzaman Siddiqui, Naseeruddin Shah, Kaifi Azmi, Habib Tanvir, Ishrat Hussain Usmani, Ghaus Bakhsh Bizenjo, Jan Nisar Akhtar, Javed Akhtar, S. Wajid Ali, Mohammad Hamid Ansari, Dhyan Chand, Rafi Ahmed Kidwai, Raja Mahendra Pratap, Ismail Gulgee, Irfan Habib, Maulana Mohammad Ali, Raja Rao, Hamiduddin Farahi, Ishtiaq Hussain Qureshi, Mian Ghulam Jilani, Syed Wazir Hasan, Saeed Rashid, Ali Naqi Naqvi, Khawaja Nazimuddin, Muhammad Mansur Ali, Samad Rizvi, Sultan Salahuddin Owaisi, Zakir Hussain, Abdul Hafeez, Mohammad Habib, Rafi Muhammad Chaudhry, Gauhar Raza, Ali Ahmed Fazeel, Shakeel Badayuni, Mushtaq Ahmad Yusufi, Abdur Rab Nishtar, Sahib Singh Verma, Hasrat Mohani, Fazal Ilahi Chaudhry, Shan-Ul-Haq Haqqee, Lala Amarnath, Akhter Husain, Ghulam Muhammad, Asghar Wajahat, Khan Habibullah Khan, Pucadyil Ittoop John, Ishwari Prasad, Zahoor Qasim, Ghulam Mustafa Khan, Munier Chowdhury, Nazir Ahmed, Mukhtar Masood, K. Asif, Maulana Syed Muhammad Zauqi Shah, Ziauddin Ahmed, Rahi Masoom Raza, Khaleel-Ur-Rehman Azmi, Farhan Nizami, Nabakanta Barua, Salman Akhtar, Abdulaziz Sachedina, Rasheed Masood, Walter Raleigh, Sardar Bahadur Khan, Maulana Shaukat Ali, Mirza Hamid Hussain, Mushtaq Ali, Abul Lais Siddiqui, Zafar Ali Khan, Ross Masood, Moonis Raza, Taher Saifuddin, Anubhav Sinha, Shaheed Latif, Abdullah Sahib, Zafar Iqbal, Mohammed Naseeb Qureshy, Rasheed Ahmad Siddiqi, Sibte Hassan, Kazi Syed Karimuddin, Mohammad Ali Ashraf Fatmi, Arif Mohammad Khan, Sri Sadiq Ali, Harsh Narain, Mohammed Usman Arif, Mohammed Ayoob, Ali Brothers. Source: Wikipedia. Pages: 430. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: General Muhammad Ataul Gani Osmani, popularly referred to as Bonga Bir Genera...More: http://booksllc.net/?id=2055375 ... Read more