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From the reviews: "Serre's notes on groups acting on trees have appeared in various forms (all in French) over the past ten years and they have had a profound influence on the development of many areas, for example, the theory of ends of discrete groups. This fine translation is very welcome and I strongly recommend it as an introduction to an important subject. In Chapter I, which is self-contained, the pace is fairly gentle. The author proves the fundamental theorem for the special cases of free groups and tree products before dealing with the (rather difficult) proof of the general case." (A.W. Mason in Proceedings of the Edinburgh Mathematical Society 1982) ... Read more

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This is a translation of Corps Locaux, supplemented with an updated bibliography and some new exercises. This book is accessible to graduate students, and can be used as a reference source by research mathematicians in algebra and number theory. ... Read more

Customer Reviews (1)

Elegant, but with Many misprints
The mathematical content and exposition are at a high level typical of Serre.

I have not finished reading the entire book, but here are some misprints I have found that may serve as a useful warning. NB: most of these errors are Not in the 3rd French edition...

Chapter 1:

section 4, pg. 14, 2nd centered display: the ramification indices should be e_{\beta} not e_{p} in the product.

section 5, pg. 15, first formula needs to be N: I_{B}-> I_{A}, Not the other way around.

section 6, pg. 17, last sentence of first paragraph, replace the inclusion symbol $\in$ with the word "in". Clearly, f is an element of A[X] and not an element of k[X]. In the French ed. Serre correctly used "dans" and did not us the symbol $\in$.

section 7, pg. 22, in proof of Prop. 21, 2nd paragraph, 3rd sentence, replace "contain" with "contains".
4th sentence: should be, "... we must have \bar{L}_{S} = \bar{K}_{T}" not \bar{L}. [separable consequence is later, namely in the Corollary(!)]

Chapter 2:

sec 1, pg. 28:third sentence should be "one sees that E is the union of (A:xA) cosets of modules xE,...". As is in the book, the sentence does not make grammatical sense.

sec 2, pg. 29: the def. of w must carry a v' not just v, that is: w = (1/m) v' is a discrete valuation of L.

sec 3, theorem 1, (i): change K to \hat{K}; so the completion of L_i has degree n_i over the completion of K.

sec 3, exercise 1: the suggested reference should say Section 3 of Bourbaki Algebra, not 7. (going by Hermann Paris 1958 as usual)

Chapter 4:

sec 1, pg. 63, prop 3, need K' (not K) in def. of e', that is: e' = e_{L/K'}.

in the proof of prop 3, the s and t for "st, t in H" need to be italicized.

sec 2, prop 6, first line of proof: gothic beta should be gothic p, that is to each x in p^{i}_{L}

sec 3, lemma 3, last line of proof: upper case Phi is nowhere defined, need lower case phi, that is: phi'(u)....so theta and phi must coincide.

sec 3, statement of lemma 5, again phi, not Phi.

Some tips for the beginner:

- Know how localization behaves as a functor via, say Atiyah-Macdonald.

- For a clean and clear proof that separable <=> nondegenerate Tr(,) see Roman's "Field Theory" (Bourbaki uses etale algebras to get this result, a bit more than needed).

- P. Samuel's "Algebraic theory of numbers" (Dover publ. now!) has a very elegant exposition of the proof of quadratic reciprocity that is alluded to at the end of section 8. ... Read more

Product Description
Jean-Pierre Serre is Professor at the Collège de France. He has written a number of books, including "Algebraic Groups and Class Fields", "Local Fields", "Complex Semisimple Lie Algebras", "Linear Representations of Finite Groups", Collected Papers (3 volumes), and "Trees" published by Springer-Verlag. ... Read more

Customer Reviews (6)

A pleasure to read, but
This book is very elegant, a pleasure to read, but not a great textbook -- after reading you are likely not to remember anything other than having enjoyed it (this is particularly true of the proof of Dirichlet's theorem). For actually learning to work in the subject (of analytic number theory), Davenport's book Multiplicative Number Theory is VASTLY superior.

An excellent book, and a great title!
Contrary to what many Americans may think of when they hear "arithmetic", this book is not about adding and multiplying (well, at least not at a basic level).Rather, this book covers several important topics in number theory - that of quadratic forms (the "algebraic" portion of the book), L-functions associated to Dirichlet characters, and modular forms (the "analytic" portion).

The section on quadratic forms is primarily devoted to a proof of the Hasse-Minkowski theorem, which is a type of "local-global" principle.This provides a natural opportunity to introduce p-adic numbers, which have become ubiquitous in the algebraic side of number theory.The analytic section starts with basic definitions and provides a proof of Dirichlet's theorem on primes in an arithmetic progression, which provides a natural opportunity to introduce an initial theory of L-functions.The last section provides an excellent introduction to the basic properties and theorems on modular forms.

What's the common theme in this book?We have the opportunity to see the origins of essential ideas in modern number theory (local-global principle, modular forms, L-functions) in a natural way.On the one hand, this book (as the other reviewers have mentioned) can be quite difficult for the beginner to read, since the exposition is sparse (but superb) and the material quite non-trivial.On the other hand, modern number theory (and much of modern mathematics) is incredibly abstract and advanced, and to see these ideas presented by a master is perhaps the best way to be introduced to the modern practice of number theory.Ultimately, with respect to difficulty it doesn't matter who or what you read to learn number theory.The subject will always be daunting to all but the most talented students, but any effort you put into reading this book is well worth it.

In summary, any aspiring number theorist (or person interested in modern number theory) needs to read this book.It can serve as an introductory text (I read it as such), but it is also an excellent reference, both for the mathematical material inside as well as the outstanding mathematical exposition.It's rare nowadays to see a mathematician of Serre's caliber also excel as an expository writer/teacher, so this book is a true gem.

not to be confused with an "arithmetic" book
In french, "arithmétique" means elementary study of rational numbers... You will not learn how to add or substract rational numbers here. An easy book to begin with this kind of maths is Andrew's: number theory (from Dover): well explained with a lot of examples...I guess that the author meant something like: "arithmetic of quadratic forms..." The truth is contained in the book's introduction: the text is made of lessons given to second-year "normaliens" from the "rue d'Ulm": those were (and still are) among the most gifted mathematic students in France; (post-graduate level...) therefore any reader without a really sharp mind and a very good level of knowledge in algebra will hardly grasp anything from the book which is really demanding; in the first ten pages, you will find a study of finite fields aimed at proving the celebrated "law of quadratic reciprocity"; it is already well beyond classical elementary arithmetic as it was taught in "classe terminale" thirty years ago and furthermore nowdays.
Then it tackles with p-adic fields, entire quadratic forms and then Dirichlet Theorem on primes in arithmetic sequences in a mere 20 pages (to set things up, this theorem is not proven in Hardy an Wright's: "introduction to the theory of numbers" because it is not an elementary theorem as mathematicians go since any proof of it requires use of advanced complex variable methods...) and at last modular forms; as you can expect, everything in the book is connected; so it looks like (it is...) an impressive piece of mathematics; it is a daunting but not impossible task to go through the whole book; let say that such hard work is rewarding and the chapter on modular forms is really a fascinating one ( and a first step towards Wiles' proof of Fermat's last theorem albeit so many more steps have to be climbed to achieve that precise aim...) ; moreover, the chapters using "analytic methods" can be read independantly; "un tour de force".

Breathtaking
Serre's work could best be summarized in one word - Elegance.
The book comprises of two distinct parts.
The first one is the 'algebraic' part. Serre's goal in this section is to give a complete classification of the quadratic forms over the rationals. As preliminaries to reaching this goal, he introduces the reader to quadratic reciprocity, p-adic fields and the Hilbert Symbol. After these three, he spends the next chapter detailing the properties of quadratic forms over Q and Q_p (the p-adic field). The reason to work over Q_p is the Hasse-Minkowski Theorem (which says that if you have a quadratic form, it has solutions in Q if and only if it has solutions in Q_p). Using Hensels Lemma, checking for solutions in Q_p is (almost) as easy as checking for solutions in Z/pZ. After doing that, he spends yet another chapter talking about the quadratic forms over the integers. (Note: the classification goal is already achieved in previous chapter).
The second half of the book is the 'analytic' one. The first chapter in this section gives a complete proof of Dirichlet's theorem while the second one studies the properties of modular forms (these are good!)
Due to the extreme elegance, the book is sometimes hard to read. This might sound like a paradox, but it's not and I'll explain why. The book takes some effort to read because it's terse and it often takes a while to figure out why something is 'obvious'. However, once you see it all, you'll realize that a great mind was guiding you through the pursuit. The choice of topics is just right to achieve the goals that the author sets out for himself. Also, I'd rather think for myself and read a smaller book than be given a huge fat tome where the author details his own thought process.
This book was my first foray into number theory and I absolutely enjoyed it. If you're considering reading it, I wish you joy in your pursuits.

Very Demanding
The book is divided into two parts -- algebraic and analytic. I've only worked through the analytic part. Anything by Serre is worth its weight in gold and this book is no exception; everything Serre covers is of the utmost importance. But Serre's style is extremely condensed and spare, and he makes no concessions to the reader in terms of motivation or examples. I can't digest more than half a page of Serre a day; however if one wants to understand the structure of a theory, Serre is ideal.

I worked through "A Course in Arithmetic" over a decade back. As I recall I covered Riemann's zeta function and the Prime Number Theorem, the proof of Dirichlet's theorem on primes in arithmetical progressions using group characters in the context of arithmetical functions, and some of the basic theory of modular functions. All of this material is also covered in Apostol's two books on analytic number theory ("Introduction to Analytic Number Theory", and "Dirichlet Series and Modular Functions in Number Theory"); Apostol goes further than Serre in the analytic part -- which is only to be expected since he is devoting two whole texts to the subject. ... Read more

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A translation from the French "Groupes Algebriques et Corps de Classes". The author, who also wrote "Galois Cohomology", presents local class field theory from the cohomological point of view. ... Read more

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From the reviews of Vol. IV:

"This is the fourth volume of J-P. Serre's Collected Papers covering the period 1985-1998. Items, numbered 133-173, contain "the essence'' of his work from that period and are devoted to number theory, algebraic geometry, and group theory. Half of them are articles and another half are summaries of his courses in those years and letters. Most courses have never been previously published, nor proofs of the announced results. The letters reproduced, however (in particular to K. Ribet and M.-F. Vignéras), provide indications of some of those proofs. Also included is an interview with J-P. Serre from 1986, revealing his views on mathematics (with the stress upon its integrity) and his own mathematical activity. The volume ends with Notes which complete the text by reporting recent progress and occasionally correct it.

The volume is an indispensable addition to previous ones and the whole set presents a fine piece of modern mathematics."

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In this softcover edition of volume IV, two recently published articles have been added, one on the life and works of Andrè Weil, the other one on Finite Subgroups of Lie Groups.

Product Description
These notes, already well known in their original French edition, give the basic theory of semisimple Lie algebras over the complex numbers including the basic classification theorem. The author begins with a summary of the general properties of nilpotent, solvable, and semisimple Lie algebras. Subsequent chapters introduce Cartan subalgebras, root systems, and representation theory. The theory is illustrated by using the example of sln; in particular, the representation theory of sl2 is completely worked out. The last chapter discusses the connection between Lie algebras and Lie groups, and is intended to guide the reader towards further study. ... Read more

Customer Reviews (2)

An encyclopedia
This book is really dense, and not recommended for the first encounter with Lie algebras. However, it contains exactly the things one ends up remembering after a year's course, so it's great for review, reminder, or a reference book. The exposition, as usual for Serre, is very clear.

One of the most valuable expositions in Lie-theory
This book is intended as a short concise overview of the theory of complex semisimple Lie algebras. Inspite of its small volume, this text is far from being of easy lecture, since it assumes the knowledge of some basic facts concerning Lie algebras, as well as associative algebras. Indeed the first chapters are a résumé, without proofs, of some basic theorems of Lie algebras. This concerns solvable and nilpotent Lie algebras, as well as some generic results on semisimple algebras (results that do not involve Cartan subalgebras).
The proper exposition begins with the third chapter, dealing with Cartan subalgebras. Two fundamental facts are exposed in this chapter: existence and conjugacy of these subalgebras. The existence is proved by exhibiting the classical construction by means of regular elements, i.e., elements of the algebra whose annihilator is of minimal dimension. The conjugacy of Cartan subalgebras, which enables us to define the numeric invariant called rank, is developed in analogous way to the book of Chevalley [Théorie des Groupes de Lie, 1951]. Chapter four is devoted to the study of the complex simple Lie algebra of rank one, sl(2,C). This algebra plays the key role in the study of semisimple algebras and their representations, which justifies a separated treatment. The irreducible representations of sl(2,C) are obtained.
The root theory is introduced in the following chapter. Here the first innovation is made, namely, developing the root systems before dealing with the Cartan decomposition. In particular, no inner product has been used yet. Root systems are defined over a real vector space V, and the Weyl group is defined as the group generated by certain involutions associated to the roots [one will observe observe the similarity of this definition and the theory of Coxeter groups]. The inner product on V is obtained as an inner product which is invariant under the Weyl group. Bases of roots and their elementary properties are developed, and how to go from a basis to another by emans of the Weyl group [it is supposed that the root system is reduced, for nonreduced systems see for example the sixth chapter of Bourbaki: Algèbres de Lie, Hermann 1967]. Then it follows the notion of Cartan matrix (obtained from the inner product previously defined), and the associated Dynkin diagram. All admissible Dynkin diagrams are enumerated, and their corresponding root systems enumerated. Chapter six begin with the classical Weyl theorems, and the Cartan decomposition of a semisimple Lie algebra is obtained. From this the root system associated to the algebra follows naturally. The core of the chapter is the existence and uniqueness proofs of semisimple Lie algebras corresponding to a root system. As an appendix, a theorem showing how to construct semisimple Lie algebras from root systems by means of generators and relations [that is, using presentations]. This result is of extreme importance, and constitutes one of the germs that lead to the notion of Kac-Moody algebras in 1968. The next chapter is a standard treatment of representation theory of semisimple Lie algebras. The existence of dominant weights is shown, from which the (canonical) bijection between dominant integral forms and the finite dimensional irreducible modules follows. The Weyl character formula is also presented, but without proof.
The final chapter presents some important results of compact groups, intended to facilitate the lecture of more advanced texts [like the book of Pontryaguin, for example].
Resuming, an excellent text that presents a good insight to the theory of complex semiimple algebras. It should however be said that probably this book is not convenient for a first contact with these structures, due to its comprised presentation and the complete absence of exercises or many examples [this applies at least to the original french edition]. Actually, some acquaitance with Lie theory is implictly supposed through the text. ... Read more

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From the reviews of Vols. I-III:

"Since their publication in 1986 J-P. Serre's Collected Papers have already become one of the classical references in mathematical research. This is on the one hand due to the completeness of the collection (132 items) and on the other, of course, due to the beautiful and clear expositions of Serre's papers and their influence on mathematics.

As listed in the preface, the three volumes cover almost all articles published in mathematical journals between 1949 and 1984, the summaries of the author's courses at the Collège de France since 1956, some of his Séminaire notes, and some items not previously published. [...]

The author's notes at the end of each volume giving corrections and important recent progress as well as improvements of the main results represent a highlight of this collection. The mathematical community definitely looks forward to further volume(s) of Serre's outstanding work."

Zentralblatt MATH

Product Description
This book reproduces J-P. Serre's 1964 Harvard lectures. The aim is to introduce the reader to the "Lie dictionary": Lie algebras and Lie groups. Special features of the presentation are its emphasis on formal groups (in the Lie group part) and the use of analytic manifolds on p-adic fields. Some knowledge of algebra and calculus is required of the reader, but the text is easily accessible to graduate students, and to mathematicians at large. ... Read more

Customer Reviews (1)

Nice and complete, but very Bourbaki-looking.
The same problem with all Bourbaki authors: They treat the subject in a very concise, abstract, and authoritative way, but present almost no motivation to introduce the subject, and they are not so used to giveextensive and accurate references. Of course Serre is a leading expert inthe field, but he (nobody) cannot be regarded as the inventor of thetheory, so the absence of such a bibliography is not justifiable.

Thecontents of the book are: Lie algebras, filtered groups and lie algebras,universal algebra of a Lie algebra, free Lie algebras, nilpotent andsolvable Lie algebras, semisimple Lie algebras, representations of sl_n,complete fields, analytic functions, analytic manifolds, analytic groups,Lie theory. Includes excercises.

Useful for graduate students and workingmathematicians, along with a "lighter" reference.

Please checkmy other reviews (just click on my name above). ... Read more

Product Description
This is an English translation of the now classic "Algbre Locale - Multiplicits" originally published by Springer as LNM 11. It gives a short account of the main theorems of commutative algebra, with emphasis on modules, homological methods and intersection multiplicities. Many modifications to the original French text have been made for this English edition, making the text easier to read, without changing its intended informal character. ... Read more

Customer Reviews (1)

Serre as usual
This book is crystal clear and reads like a bedtime story.I wish I had it when I took commutative algebra. ... Read more

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This extraordinary volume contains a large part of the mathematical correspondence between A. Grothendieck and J-P. Serre. It forms a vivid introduction to the development of algebraic geometry during the years 1955-1965. During this period, algebraic geometry went through a remarkable transformation, and Grothendieck and Serre were among central figures in this process.In the book, the reader can follow the creation of some of the most important notions of modern mathematics. The letters also reflect the mathematical and political atmosphere of this period. They are supplemented by J-P. Serre's notes, which give explanations, corrections, and references to further results.The book is a unique bilingual (French and English) volume. The original French text is supplemented here by the English translation, with French text printed on the left-hand pages and the corresponding English text printed on the right. The book also includes several facsimiles of original letters. The original French volume was edited by Pierre Colmez and J-P. Serre. The English translation for this volume was prepared by Catriona Maclean with the assistance of Leila Schneps and J-P. Serre.The book should be useful to specialists in algebraic geometry, mathematical historians, and to all mathematicians who want to experience the unfolding of great mathematics. ... Read more

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This classic book contains an introduction to systems of l-adic representations, a topic of great importance in number theory and algebraic geometry, as reflected by the spectacular recent developments on the Taniyama-Weil conjecture and Fermat's Last Theorem. The initial chapters are devoted to the Abelian case (complex multiplication), where one finds a nice correspondence between the l-adic representations and the linear representations of some algebraic groups (now called Taniyama groups). The last chapter handles the case of elliptic curves with no complex multiplication, the main result of which is that the image of the Galois group (in the corresponding l-adic representation) is "large." ... Read more

Customer Reviews (2)

Serre is great
This book, as all books by Serre, is very good.He writes with precision, always giving the reader the central ideas.I can only think of Milnor as an author to compare to him.Buy the book if you are interested at all in the material.I am buying all books I can get by Serre.

excellent
great reference if you are doing work in algebraic representations of geometric structures at the graduate level.There aren't very many exercises, but as a graduate reference text, exercises don't do a whole lot of good.great explanations, and clear proofs. ... Read more

Product Description
This edition reproduces the 2nd corrected printing of the 3rd edition of the now classic notes by Professor Serre, long established as one of the standard introductory texts on local algebra. Referring for background notions to Bourbaki's "Commutative Algebra" (English edition Springer-Verlag 1988), the book focusses on the various dimension theories and theorems on mulitplicities of intersections with the Cartan-Eilenberg functor Tor as the central concept. The main results are the decomposition theorems, theorems of Cohen-Seidenberg, the normalisation of rings of polynomials, dimension (in the sense of Krull) and characteristic polynomials (in the sense of Hilbert-Samuel). ... Read more

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This new edition includes a survey (mostly without proofs) of the main results obtained in the 30 years following original publication. It also incorporates newer material, including two "résumés de cours" at the Collège de France (1990 - 1991 and 1991 - 1992), and an updated bibliography.

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From the reviews of Vols. I-III:

"Since their publication in 1986 J-P. Serre's Collected Papers have already become one of the classical references in mathematical research. This is on the one hand due to the completeness of the collection (132 items) and on the other, of course, due to the beautiful and clear expositions of Serre's papers and their influence on mathematics.

As listed in the preface, the three volumes cover almost all articles published in mathematical journals between 1949 and 1984, the summaries of the author's courses at the Collège de France since 1956, some of his Séminaire notes, and some items not previously published. [...]

The author's notes at the end of each volume giving corrections and important recent progress as well as improvements of the main results represent a highlight of this collection. The mathematical community definitely looks forward to further volume(s) of Serre's outstanding work."

Zentralblatt MATH

Customer Reviews (1)

WARNING: this does not come in the advertised size
The actual size of this book is roughly 10 x 7 inches, about half an inch longer and wider than the other three volumes on amazon (it also does not have the folded cover pages)! Confusingly, it does have the same ISBN as the volume of the advertised size (available elsewhere). ... Read more