Reviews (13)

The 'Queen Bee' of Basic Algebra Texts
I have no doubt that Lang's `Algebra' will go down recorded (if not already) as a great achievement in modern mathematics and its literature. I strongly believe that Lang, instead of just writing another introduction, tries to present material which he considers introductory in a more mature level that many people take for granted, while those who don't truly appreciate his unique style and clarity.

I cannot think of any text in mathematics that so admirably comprises so much material in a way that gets `straight to the heart' of the matter with no unnecessary `fluff' in between (Serre, Atiyah and Milnor's books are my only exceptions). Every chapter is supplied with plenty motivations and references that I find essential for my research. The exercises are highly instructive and provide different directions for further investigation that no other text can match in either breadth or literature.

As an example, I'd like to mention his treatise in Galois Theory. The fact that he states the heart of Galois Theory (the Fundamental Theorem of Galois Theory) right at the beginning is almost vital in presenting the theory. (This is not so in other mathematical theories like zeta functions where the heart is more the Fourier Transform).Moreover, Kummer Theory, Galois Cohomology, Infinite Galois Theory, the `Inverse Problem' (just to name a few) are also discussed in the chapter in considerable detail to whet the reader's appetite and give the she/him a sense of direction as to where the theories seem to be heading. He also provides very instructive examples of computing galois groups.

I must agree, however that the book is poorly indexed. This is understandable, however, because the table of contents and layout of the text are sufficient for any reader to find what she/he is looking for.

Lang also dedicates three chapters of algebra `tuned' for the first three chapters of Hartshorne's `Algebraic Geometry' (an essential book for almost any aspiring mathematician)..

In addition, throughout the book, Lang provides a historical overview of the topic in discussion with many references, which is necessary, for example, in homological algebra (e.g. Grothendieck's Tohoku paper).

I for one am truly grateful for what Serge Lang has done for mathematics and its community. He alone may be regarded as a Euclid of the 20th Century. He more than once gave students and mathematicians access to new topics of mathematics that lacked proper treatise.
It is not surprising that one finds the name Lang all over experts' references and list of favorite books.

(For the frustrated beginner, however, I recommend a Isaacs, Mac Lane and Birkhoff, Herstein or Jacobson I as an introduction.)

Poor Text, Poor Reference
The only thing that comes through cleary in this book is the author's arrogance.The reader will get a very good sense that Lang understands the material, but the reader will also get the sense that there must be a better book out there.Even as a reference, this book is lacking.The index is incomplete, and many important concepts are either poorly defined or the relevant equivalent definitions are not given.Finally, the order in which the material is presented is haphazard at best.This is truly a text only for those with a Ph.D. who work primarily in the field of Algebra.

This will teach you how to run if you know how to walk
Lang's algebra book is one of the best algebra books available today. I agree with what most other readers have said. Namely, this shouldn't be your first foray into the subject, the proofs are often terse and take a good amount of time to absorb and there is a conspicuous lack/obscurity of examples. To cite an example, he gives a non-singular projective group variety as an example of a certain group. I shall not give an example of a terse proof. Let's just say that it suffices to note that whenever he says something is 'obvious', the non-expert reader should be prepared to scribble on 4-5 sheets of paper if she wishes to understand why it's 'obvious'.

The core matter (groups, rings, fields, modules) is the same as that you'd find in any other book. As far as topics are concerned, there are just too many fascinating topics in Algebra to cover in one book - even in one like Lang. He covers a fairly wide assortment of topics though. For instance, he covers most of the commutative algebra one would find in Atiyah-Macdonald. He also has a chapter and half on Algebraic Geometry which provides a good preparation for a treatment of schemes like that in Hartshorne Chapter 2,3. His section on Galois theory is detailed and even gets into Galois Cohomology. His chapter on Valuations gets into the theory of Local Fields, but only just. The chapters on multilinear algebra and representation theory are fairly detailed. I talk about the section on Homological Algebra later.

Regarding category theory, Lang likes to phrase his definitions in the language of category theory for a reason. It's much much better this way. Category theory is an elegant way of describing some commonly occuring themes in Mathematics, particularly algebra. His preliminary section on category theory provides a good foundation to study the rest of his book. Another advantage of using category theory is that this prepares the reader well for further study in Algebraic Geometry and Algebraic Number Theory where the language of category theory is ubiquitous. On a related note, the book contains all the homological algebra necessary to read Hartshorne's Algebraic Geometry which is indeed quite wonderful for the reader who's not prepared to fight through Eisenbud's encyclopedia on commutative algebra.

Reviews (3)

dense and uninviting
This is a typical mathematical monograph
which means it is densely written with
almost no examples.It's too bad since
that makes decoding the text much more
timeconsuming.

There is a lot here for such a short book
This book is a pretty good introduction to the theory of Lie algebras and their representations, and its importance cannot be overstated, due to the myriads of applications of Lie algebras to physics, engineering, and computer graphics. The subject can be abstract, and may at first seem to have minimal applicability to beginners, but after one gets accustomed to thinking in terms of the representations of Lie algebras, the resulting matrix operations seem perfectly natural (and this is usually the approach taken by physicists). The book is aimed at an audience of mathematicians, and there is a lot of material covered, in spite of the size of the book. Readers who desire an historical approach should probably supplement their reading with other sources. Readers are expected to have a strong background in linear and abstract algebra, and the book as a textbook is geared toward graduate students in mathematics. Only semisimple Lie algebras over algebraically closed fields are considered, so readers interested in Lie algebras over prime characteristic or infinite-dimensional Lie algebras (such as arise in high energy physics), will have to look elsewhere. Physicists can profit from the reading of this book but close attention to detail will be required.

The first chapter covers the basic definitions of Lie algebras and the algebraic properties of Lie algebras. No historical motivation is given, such as the connection of the theory with Lie groups, and Lie algebras are defined as vector spaces over fields, and not in the general setting of modules over a commutative ring. The four classical Lie algebras are defined, namely the special linear, symplectic, and orthogonal algebras. The physicist reader should pay attention to the (short) discussion on Lie algebras of derivations, given its connection to the adjoint representation and its importance in applications. The important notions of solvability and nilpotency are covered in fairly good detail. Engel's theorem, which essentially says that if all elements of a Lie algebra are nilpotent under the 'bracket", then the Lie algebra itself is nilpotent, is proven.

The second chapter gives more into the structure of semisimple Lie algebras with the first result being the solution of the "eigenvalue" problem for solvable subalgebras of gl(V), where V is finite-dimensional. Cartan's criterion, giving conditions for the solvability of a Lie algebra, is proven, along with the criterion of semisimplicity using the Killing form. The representation theory of Lie algebras is begun in this chapter, with proof of Weyl's theorem. This theorem is essentially a generalization to Lie algebras of a similar result from elementary linear algebra, namely the Jordan decomposition of matrices. Again, physicist readers should pay close attention to the details of the discussion on root space decompositions.

This is followed in chapter 3 by an in-depth treatment of root systems, wherein a positive-definite symmetric bilinear form is chosen on a fixed Euclidean space. These root systems enable a more transparent approach to the representation theory of Lie algebras. The theory of weights along with the Weyl group, allow a description of the representation theory that depends only on the root system. In addition, one can prove that two semisimple Lie algebras with the same root system are isomorphic, as is done in the next chapter. More precisely, it is shown that a semisimple Lie algebra and a maximal toral subalgebra is determined up to isomorphism by its root system. These maximal toral subalgebras are conjugate under the automorphisms of the Lie algebra. The author further shows that for an arbitary Lie algebra that is true, if one replaces the maximal toral subalgebra by a Cartan subalgebra. The proofs given do not use algebraic geometry, and so they are more accessible to beginning students.

In chapter 5, the author introduces the universal enveloping algebra, and proves the Poincare-Birkhoff-Witt theorem. The goal of the author is to find a presentation of a semisimple Lie algebra over a field of characteristic 0 by generators and relations which depend only on the root system. This will show that a semisimple Lie algebra is completely determined by its root system (even if it is infinite dimensional).

Chapter 6 is very demanding, and will require a lot of time to get through for the newcomer to the representation theory of Lie algebras. Weight spaces and maximal vectors are introduced in the context of modules over semisimple Lie algebras L. Finite dimensional irreducible L-modules are studied by first considering L-modules generated by a maximal vector. It is shown that if two standard cyclic modules of highest weight are irreducible, then they are isomorphic. The existence of a finite dimensional irreducible standard cyclic module is shown. Freudenthal's formula, which gives a formula for the multiplicity of an element of an irreducible L-module of heighest weight, is proven. A consideration of characters on infinite-dimensional modules leads to a proof of Weyl's formulas on characters of finite dimensional modules.

The last chapter of the book considers Chevelley algebras and groups. Their introduction is done in the context of constructing irreducible integral representations of semisimple Lie algebras.

Excellent Introduction to Lie Algebras
Humphreys' book on Lie algebras is rightly considered the standard text.Very thorough, covering the essential classical algebras, basic results on nilpotent and solvable Lie algebras, classification, etc. up to andincluding representations.Don't let the relatively small number of pagesfool you; the book is quite dense, and so even covering the first 30 pagesis a nice accomplishment for a student.Small caveat, the notation mightbe a bit confusing until you get used to it, but this is a common problemdue to having both a Lie and a matrix product floating around, and is not afault of the text.There is also a nice selection of exercises, between 5and 10 per section.

Isbn: 3540426507
Sales Rank: 568654
Subjects:  1. Algebra - Linear    2. Geometry - Algebraic    3. Group Theory    4. Lie Groups    5. Lie algebras    6. Mathematics    7. Science/Mathematics    8. Topological Groups    9. BN-pairs    10. Coxeter groups    11. Mathematics / Group Theory    12. Tits systems    13. Weyl groups    14. root systems    15. semi-simple Lie algebras   

Reviews (3)

A baptism of fire for Algebraic Geometry
Some people believe that, for getting into algebraic geometry (by this I mean Grothendieck-like AG, with schemes and all that), one needs a monolithic training in commutative algebra (something like both volumes of Zariski-Samuel, for example). I disagree. This little book seems to be specially suited to those who want to learn AG. It's a bit too brisk, specially at the beginning - if you don't already have an acquaintance with the basics of groups, rings and ideals, you may run into trouble - but veryilluminating. Masterful choice of topics, great exercises (as a matter of fact, about half the topics of the book, and more specifically the ones that are directly related to AG, are treated in the exercises, some of them quite challenging) - like one said before, it looks like a "chapter 0" of Hartshorne's book on AG. The authors consciously estabilish relations between the commutative algebra and the modern foundations of AG over and over along the way, illuminating both topics.

For the algebra itself, it also gets on well with Rotman's "Galois Theory" and MacDonald's out-of-print introduction to AG, "Algebraic Geometry - Introduction to Schemes", besides being the perfect preamble in commutative algebra to the books of Mumford and Hartshorne. A gem.

A gem of mathematical writing
This is how mathematics texts SHOULD be written. As in technical writing, the smaller text is the better written text. Everything is clean and direct, with clairity obviously a prime consideration. One never gets mireddown. The proofs are always as close to a "THE BOOK" proof aspossible, with illuminating examples, and plenty of excercises, many withoutlines for solution, which makes the book ideal for self study. This bookis a revelation. If I had to take only one math text with me to a desertisland, this would be the one.

Reviews (3)

Crystal clear overview of a traditionally abstract subject
The theory of schemes is usually thought to be highly abstract and esoteric, and one that makes the study of algebraic geometry even more difficult. The authors definitely dispel this notion in this book, which could have been called "A Concrete Introduction to Schemes", because of the clarity with which the concepts are introduced and explained. After studying this book, one will understand and appreciate the power of schemes in algebraic geometry. The authors do an even better jobthan they did in their earlier and short work "Schemes: The Language of Modern Algebraic Geometry", which is now out of print.

In chapter 1, the main definitions are given and the basic concepts behind schemes outlined. That schemes are more complicated than varieties is readily apparent even in this beginning chapter, where they are thought of as corresponding to the spectrum of a commutative ring with identity. Very elementary exercises are given to help the reader gain confidence in the constructions involved. They authors do have to discuss some sheaf theory, but they show its relevance nicely in this chapter. They also discuss the notion of a fibered product as a generalization of the idea of a preimage of a set under the application of a function and relate it to the construction of the functor of points. The role of the functor of points as reducing schemes to a kind of set theory is brought out beautifully here.

The next chapter gives many examples of schemes, with the first examples being reduced schemes over algebraically closed fields, these being essentially the ordinary varieties of classical algebraic geometry. The authors then give examples of schemes, the local schemes, which are more general than varieties. When departing from the assumption of a field that is not finitely generated, extra points will have to be added to classical varieties. The fact that only one closed point appears is compared to the case of complex manifolds, via the concept of a germ. This is a very helpful comparison, and one that further solidifies the understanding of a scheme in the mind of the reader. The authors give the reader a short peek at the etale topology in one of the examples. Examples are then given where the field is not algebraically closed, generalizing classical number theory, and non-reduced schemes, where nilpotents are present. The chapter ends with examples of arithmetic schemes where the spectra of rings are finitely generated over the integers.

Projective schemes are the subject of Chapter 3, and are defined in terms of graded algebras and invariants of projective schemes embedded in projective space are discussed. The Grasmannian scheme is discussed in detail as an example of a projective scheme. Interestingly, Bezout's theorem, very familiar from elementary algebraic geometry, is generalized here to projective schemes.

Constructions from classical algebraic geometry are generalized to schemes in Chapter 4. The first one discussed is the notion of a flex, which deals (classically) with the locus of tangent lines to a variety. The flexes are defined in terms of the Hessian of the variety, the latter being generalized by the authors to define a scheme of flexes. The notion of blowing up is also generalized to the scheme setting, with the authors motivating the discussion by blowing up the plane. The discussion of blow-ups along non-reduced subschemes of a scheme and blow-ups of arithmetic schemes is fascinating and the presentation is crystal clear. Fano varieties are also generalized to Fano schemes in the chapter. Most of the information about these schemes are contained in the exercises, and some of these need to be worked out for a thorough understanding.

The next chapter is more categorical in nature, and deals with generalizations of the classical Sylvester construction of resultants and discriminants to the scheme setting.

In the last chapter the authors return to the functor of points, and motivate the discussion by asking for a parametrization of families of schemes. The authors show, interestingly, that using the functor of points one can more easily compute geometric information about a scheme than using its equations. They illustrate this for the Zariski tangent space. Then after an overview of Hilbert schemes they close the book by introducing the reader to moduli spaces and a hint of algebraic stacks. No end in sight for this beautiful subject..........

A very good start
This book is clear, well written, and has a nice balance of generalities and examples. If you know the basics of rings and modules, this book will show you what schemes are and why they are useful for several different problems: for example, number theory, or studying singularities. I find it a helpful companion to Hartshorne's ALGEBRAIC GEOMETRY. But this book does not get to cohomology, and so cannot actually get to the working methods in the subject. For that, you need Hartshorne.

Reviews (6)

The Clearest Introduction to Algebraic Geometry
Hartshorne is by far the most clear and compact of all introductions to algebraic geometry. I agree with what most of the other reviewers have said about the book. It is absolutely amazing how the Grothendieck et al. approach has provided us with tools that we almost take for granted, whereby everything seems precise and crystal clear. AG has definitely made a profound influence on our philosophy of mathematics.

Lee Carlson is absolutely right about exploring the literature for geometrical motivation and different perspectives. Throughout his book, Hartshorne does not reference enough to the Italian and Weil literature. (However, he does inform his reader about this in the intr.).

Here are some classics for different homological perspectives (which as he mentions are essential for the budding algebraic geometer to read)

Godement-Topologie Algebrique et Theorie des Faiscieux(???)
Serre- Faisceux Algebrique Coherents (Oeuvres I)
Grothendieck's Tohoku paper (send me an e-mail and I'll send it to you)

All of them are definitely still worth reading and are probably the best introductions to sheaves and categories. In fact, at the moment I'm reading Serre's FAC and am having a wonderful time. If you don't know French, then learn it! The French is very easy and all a non-french speaker will probably need is a small dictionary.

It is my opinion that one should should explore the literature as soon as possible and not get bogged down by the countless no. of other introductions. Algebraic geometry is a huge and central subject that one cannot afford to waste time with the basics.

To find other modern classics, I think Hartshorne's reference is superb. If you really want go back in time (but not too back), then check out in particular Jacobi's CW and Baker's Abelian Functions : Abel's Theorem and the Allied Theory of Theta Functions (Cambridge Mathematical Library). (Obviously the litarature abounds in both directions of the 19th Century)

A great overview and reference to the 20th Century history of algebraic geometry is Dieudonne's little book `History of Algebraic Geometry' (worth the rediculus pricing). In particular, Serre's philosophy in introducing cohomology is discussed in the book, which I found very interesting.

Great further reads are (leaving the arithmetic story [e.g. Elliptic Curves] aside) :

SGA, EGA (in french too!)

The 1974 AMS Arcata, which shows a lot of the directions AG is heading.

Lang or Mumford- Abelian Varieties (both are great.
Mumford uses schemes, but Lang touches arithmetic).

Milne- Etale Cohomology.
This is a very good text that will lead the reader to Deligne's Conjecture de Weil I & II.

Serre- Algebraic Groups and Class Fields.
Contains different approaches to the Riemann-Roch Thm and discusses goemetric class field theory.

Griffiths and Harris.
This gives a different approach to algebraic geometry. It's more geometric and touches upon many concepts that originated from (introduced by) de Rham and E. Cartan (e.g. Currents, differential forms etc.)

Mumford- Lectures on algebraic surfaces.
Shafarevitch et al.- Algebraic surfaces.
These are special topics and the books are 'raw' in the sense that they are 'straight from the horses' mouths'.

(...)

The reader should also be aware of the new and surprising relationaships AG is having with the world (lee carlson's review gives an idea) and number theory. In addition, one of the two 2002 Fields medalists (V. Veovodsky) got this recognition for his deep and insightful interplay between Algebriac Geometry (motives) and Algebraic topology (homotopy, K-theory).

Hartshorne's exercises are extremely instructive to the point of absolutely crucial because of the scarcity of examples in the book.
For those self-studying, a Google search will probably provide one with solutions to some of the exercises in the chapters.

THE book for the Grothendieck approach
This is THE book to use if you're interested in learning algebraic geometry via the language of schemes.Certainly, this is a difficult book; even more so because many important results are left as exercises.But reading through this book and completing all the exercises will give you most of the background you need to get into the cutting edge of AG.This is exactly how my advisor prepares his students, and how his advisor prepared him, and it seems to work.

Some helpful suggestions from my experience with this book:
1) if you want more concrete examples of schemes, take a look at Eisenbud and Harris, The Geometry of Schemes;
2) if you prefer a more analytic approach (via Riemann surfaces), Griffiths and Harris is worth checking out, though it lacks exercises.

Experiences of a rather below-average graduate student
(that's me.)

I agree with the other reviewers' comments concerning the phenomenal depth and breadth of the topics covered in this book.Hartshorne builds the soaring edifice of modern algebraic geometry from the ground up.All the way through, the exposition is concise and absolutely clear.The proofs strike an excellent balance between meticulousness and readability.

The approach he takes seems to be to try to acquaint the reader with as much formalism as possible as quickly as possible, and he seems reluctant to offer any sneak previews of vital concepts such as divisors, differentials, and flatness until the reader's brain is "ripe".As a result, Hartshorne is able to state and prove results under extremely general hypotheses.This approach also benefits the kind of reader who wishes to use this as a reference book.

It's important also to note the disadvantages of Hartshorne's approach:Time and again, I found myself utterly baffled by the definitions, because the motivations for them are lacking.

To give a minor example, take the definition (in chapter 1, part 3) of a morphism between two varietes.First, regular functions from a variety over k to k are defined as those that are locally representable as quotients of polynomials (without bothering to give an example of a case of a regular function for which more than one such representation is needed).Then a morphism f: X -> Y is defined as a Zariski-continuous function with the property that whenever you have an open subset V of Y, and a regular function V -> k, then f^-1(V) -> V -> k is regular.There's nothing wrong with this definition, of course, but I found it very difficult to make sense of, initially.A morphism, after all, is supposed to be something that preserves structure, but it's not immediately obvious what "structure" is being preserved in this case (and the full details of this aren't spelt out until much later, after sheaves have been defined).A better didactic approach, I think, would be either (1) to define morphisms of affine varieties simply as functions given by polynomials, and then show that the above definition is the only natural way of generalising this, or (2) to briefly introduce sheaves at the outset, making it clear that the "structure" we wish to define on a variety consists precisely of the sheaf of regular functions.

Another negative effect of Hartshorne's approach is that, if you have to traverse a mire of formalism before meeting an idea, it makes the idea seem more complicated than it actually is.

Certainly there's nothing to stop a dedicated reader just ignoring any temporary befuddlements, secure in the knowledge that eventually everything will make sense, but not all of us have the patience.This book contains an almost ridiculous number of exercises - most of which are supposed to be "formalities", there to flesh out the definitions, but many contain absolutely crucial definitions and lemmas.Attempting to do all the exercises as you go along is very taxing work indeed, and becomes demoralising whenever you get stuck.Perhaps the best strategy is to do only those exercises that are interesting or important for later work.
Also, as others have noted, this book is very tough going on those who don't already have some familiarity with commutative algebra and (later on) homological algebra.

Isbn: 0521786754
Sales Rank: 414795
Subjects:  1. Finite Mathematics    2. Finite groups    3. General    4. Group Theory    5. Mathematics    6. Science/Mathematics    7. Groups & group theory    8. Mathematics / Algebra / General   

Isbn: 0486666514
Sales Rank: 373784
Subjects:  1. Algebra - General    2. Algebraic fields    3. Field extensions (Mathematics)    4. Mathematics    5. Science/Mathematics   

Reviews (23)

Et Tu, Bah
Having read, My Cousin, My Gastroenterologist, I had high hopes for Et Tu, Babe. However, other than a few hilarious venues(the "everything" sandwich, visceral tattoos, and the Schwarzeneggerization of America), this was a disappointment. I wound up skimming through too much tedium. I suspect Leyner had personal problems midway (the Arlene Scene) as the 2nd half of the bookd r a g s. Nun thee less, Leyner when he's on is one of the funniest orgasmic writers around.

Brilliant
It's a Mark Leyner book. That's about as much of a compliment as I can think of. Really a uniquely disturbing individual. But a brilliant, innovative writer.

Reviews (7)

Not the best for absolute beginners, but great reference
For starting out in Latex, I recommend one of the many free online tutorials, such as the "not-so-short introduction to Latex".That will get you more than started.This book is a great add-on reference to that, and the level is just right: not too wordy, but easy to follow.It has much deeper coverage of how to do graphics and drawings, how to create a bibliography, how to create postscript and pdf files, etc, in addition to the basics, of course.

Brief, effective introduction to Latex
This is an ideal introduction to the TeX line of text formatting tools. It starts with a high level overview of the toolset. Then dives into document construction, then into tables, lists, mathematical formatting, and into user constructions and extensions for bibliographies, as well as other topics. PDF and Postscript formatting is covered. As well as AMS-TeX which is more extensions or mathematical formatting.

On the downside the substance of the text is only two-thirds of the book. Appendices make up the rest. Which necessarily means that the coverage is a little more terse than you would expect for a book this size.

That being said, this is still an excellent introduction to the TeX set of tools for anyone who is just starting out.