Book Description
This text offers a clear, efficient exposition of Galois Theory with complete proofs and exercises. Topics include: cubic and quartic formulas; Fundamental Theory of Galois Theory; insolvability of the quintic; Galois's Great Theorem (solvability by radicals of a polynomial is equivalent to solvability of its Galois Group); and computation of Galois groups of cubics and quartics. There are appendices on group theory, ruler-compass constructions, and the early history of Galois Theory. This book provides a concise introduction to Galois Theory suitable for first-year graduate students, either as a text for a course or for study outside the classroom. This new edition has been completely rewritten in an attempt to make proofs clearer by providing more details. The book now begins with a short section on symmetry groups of polygons in the plane, for there is an analogy between polygons and their symmetry groups and polynomials and their Galois groups; this analogy can serve as a guide by helping readers organize the various field theoretic definitions and constructions. The exposition has been reorganized so that the discussion of solvability by radicals now appears later and several new theorems not found in the first edition are included (e.g., Casus Irreducibilis).
Customer Reviews:
A good book for a first date with Artin's version of Galois theory.......2007-07-16
I came to know this book in its first edition and I particularly prefer that one to this second. The contents in this new addition has been expanded a bit, new worked out examples has been added and new details has been given to some proofs, but in my opinion the book has lost some of the lucidity that it claims to have in the preface.
Despite this the book continues being one of the best introduction to Galois theory and I recommend it to anyone pursuing this subject, even abstract algebra itself, for the first time, since it gathers all the elemetary material in a succint form.
But if you really want to get a feeling of the beautiful ideas that ocurred to Galois you must go first to his original writings ("read the Masters" as Edwards recommends), and then come to this book to appreciate the value of the generality of Artin's version of Galois theory in terms of field automorphisms.
Good Reference.......2003-03-09
This text serves as a good reference, however, it is not necessarily well-suited for self-learning. The exercise sets need some revamping.
Sweet and Concise.......2002-07-28
I used this book as my course text book in my Galois Theory course in my undergraduate. It is pretty self-contained, so even if you forget some of Group and Fields Theory, you will still find it very readable. The little book guides to the Great Theorem by Galois (f is solvable by radical if and only if Gal(f/Q) is solvable) by breaking it into many nice sections. A drawback of this book is that some proofs at the end are omitted because they are put as execrises! Also, bear in mind that this book focus in Q and R (rational and real numbers) field only. In any case, I still recommend this book because it is sweet and concise, for which I can read it like a novel!
Classical "Artin's" Galois theory.......1997-12-08
From the definition of a conmutative ring to the fundamental theorem to solvability of equations by radicals in 65 pages, 80 theorems and 106 exercises. The exposition, wich follows the now classical tradition of "Artin's" Galois theory, is quite efficient, packing much material in a limited number of pages. The greatest asset of this book is its nice selection of topics, focusing on the fundamental theorem of Galois theory and its application to solbability of equations by radicals, but pausing to make excursions to finite fields or to work out explicitly some iluminating examples. The style is no-nonsense, crisp but nop hurried. The final 40 pages consist of appendices discussing group theory, ruler an compass constructions and old-fashioned Galois theory. This latter appendix deserves special mention, since it is not customary for a textbook of this size and scope to include such a detailed sketch of the historical motivations behind the theory it describes. One can only agree with the author when he wonders how such thoughts occurred to Galois in the late 1820's, and be grateful to him for providing his reader with material for an answer.
Book Description
This book's organizing principle is the interplay between groups and rings, where “rings” includes the ideas of modules. It contains basic definitions, complete and clear theorems (the first with brief sketches of proofs), and gives attention to the topics of algebraic geometry, computers, homology, and representations. More than merely a succession of definition-theorem-proofs, this text put results and ideas in context so that students can appreciate why a certain topic is being studied, and where definitions originate. Chapter topics include groups; commutative rings; modules; principal ideal domains; algebras; cohomology and representations; and homological algebra. For individuals interested in a self-study guide to learning advanced algebra and its related topics.
Customer Reviews:
excellent.......2006-03-30
This is a very clear introduction to graduate-level algebra. It is much better organized than Dummit and Foote. I particularly like the treatment of modules.
The worst mathematics book I have ever read!!!.......2004-07-21
I gave this book one star only because I couldn't give it a score of zero!!! Although many professors say that this book is excellent, remember they are professors who already understand the material. This book shows no examples, and the examples that it does show end abruptly with comments such as "all items are routine." Routine!!! Please show me what to do so that I don't have to spend more money on a separate study guide. Aren't mathematics texts expensive enough? This book may be an excellent addition to a professors library but this book should never, ever be used as a primary text for students.
Good for Self-Study.......2003-08-22
This is a tough book to review, because it is not clear who the real audience is supposed to be. The author says that it is aimed at first-year graduate students, with a bunch of extra material that can be referred back to during the second year and beyond. The earlier chapters also include efficient reviews (with sketched proofs) of material that should be familiar to those who have taken undergraduate algebra.
This characterization is debatable. Based on my experience reading most of the first six chapters (the first 400 out of about 1000 pages), I would say that the level of sophistication is roughly that of Dummit and Foote's "Abstract Algebra", which is usually considered an undergraduate book. D&F can sometimes be harder to read, and that is in part because Rotman's exposition is better (in my opinion), but also because D&F introduce more difficult material earlier. Whether D&F's approach is better is questionable; I find Rotman to be a much smoother read, but the organization is quite different -- for example, one does not encounter noncommutative rings until deep into the book, whereas Dummit and Foote introduce them immediately upon defining rings. On the other hand, early in the coverage of D&F's chapter on rings, one has to digest Zorn's Lemma and its applications almost from the beginning, whereas Rotman (I think wisely) pushes this back into a later section. In general, D&F introduce a lot of hairy examples that by themselves require a lot of effort to digest (thereby impeding the reader's progress through the core material), whereas Rotman's examples tend to be straightforward, at least as new concepts are being presented.
So, overall, the exposition flows more smoothly in Rotman's book, and the reader can cover the basics more quickly with less time spent on tangential examples and early generalizations. Also, Rotman's proofs are usually much cleaner and the overall style is very nice. It's more pleasant to read than Dummit and Foote. But this comes at a cost: Dummit and Foote do cover more material, and generalize at an earlier stage, than Rotman does.
But my biggest gripe concerns the exercises. Put simply, Rotman's are far too easy for what is being pitched as a graduate course. In fact, they are in general far easier than the homework problems I sweated through when I took honors undergraduate algebra. They're barely adequate to convince the reader that he has a basic grasp on the material, and there are almost no hard ones, let alone really tough, thought-provoking open-ended problems like one encounters in Herstein's "Topics in Algebra" (an undergraduate book). There are certainly no exercises in Rotman's book that would be of any use for a graduate student preparing for qualifying exams. They're not even much of a workout for a decent (honors student) undergraduate.
So, what is this book good for? I think it's great for reading material that is usually harder to understand elsewhere. Rotman has a real knack for clear mathematical exposition, and some of the chapters are a real joy to read. (Side note: there are also a lot of typos, at least in the first printing. The author maintains an errata list at his web site, and a second printing is coming soon. There are still many errata that he didn't catch, but they're fairly minor and do not detract significantly from the reading.) But this is simply not suitable for a primary graduate text or reference. Most good schools are going to demand more of their graduate students, and one is inevitably going to have to read Lang or Hungerford (and work through their exercises) to achieve competence at the graduate level. Rotman's book is a kinder, gentler book upon which to fall back when those books are inscrutable, as is all too common. I do recommend it highly for that purpose -- I think it's a very good secondary book.
An excellent Text.......2003-07-19
To begin with, don't let the title scare you. After having read through Rotman's book I am suprised that this text had not crossed my path earlier. It is a wonderful book and must have for any inspiring Algebraist. Moreover, I am quite shocked that the larger universities have not adopted this book.
(a) This book could quite easily be used as the standard third/fourth year undergraduate introduction to Abstract Algebra. In particular, the first four chapters provide a solid foundation for a moderate paced one semester course at which point the instructor has many different options for additional topics based on the performance of his/her class.
(b) Those students that move on to the graduate level, and obviously to a university using this book, would both be familiar with the temperment and flow of the author as well as devoid of the requirement of having to purchase another expensive Mathematics text. For example, my undergraduate Algebra text was Hungerford's and post completion the logical step, being familiar with his style, was to purchase Hungerford's graduate text. For those not familiar, let me tell you there is a night and day difference with repsect to how the material is presented.
(c) The remaining 7 chapters take the willing student on a pleasant tour of ring/module theory, some advanced group theory (for the inspiring group theorist I highly recommend the authors graduate text "Group Theory"), algebras(linear included), Homology(some cohomology) and finally some algebraic number theoretic concept under the heading of Commutative Rings III.
(d) Lastly, Rotamn does not get needlessly bogged down in any one section of the book. The flow is smooth, to the point with precise definitions, examples, and ample exercises.
I have only two negative remarks: one, the failure to include more aspects of field/Galois theory. This may be due to the author already having published a book entitled "Galois Theory". Two, the failure to devote an entire section to Finite Fileds and possibly some its applications. But this failure is minimal since, at present, the majority of Algebra texts, fail to adequately introduce and motivate Finite Fields.
Great Book!..........2002-10-07
I previously purchased Rotman's First Course in Abstract Algebra, and fell in love with it. So when I saw he a Second Abstract Algebra book, I had to have it. I am currently taking a Graduate Level Modern Algebra course, and I find this book to be a great help in my Studies. I wouldn't be as interested in Modern Algebra as I am now if it weren't for this book. I love this book and I would reccomend it to anyone who is interested in Modern Algebra, or taking a course in Modern or Abstract Algebra.
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An Introduction to Group Rings (Algebras and Applications, Volume 1) (Algebra and Applications)
César Polcino Milies , and
S.K. Sehgal
Manufacturer: Springer
ProductGroup: Book
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ASIN: 1402002386 |
Book Description
Group rings play a central role in the theory of representations of groups and are very interesting algebraic objects in their own right. In their study, many branches of algebra come to a rich interplay. This book takes the reader from beginning to research level and contains many topics that, so far, were only found in papers published in scientific journals and, whenever possible, offers new proofs of known results. It also includes many historical notes and some applications.
Audience: This book will be of interest to mathematicians working in the area of group rings and it serves as an introduction of the subject to graduate students.
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Introduction to Vertex Operator Superalgebras and Their Modules (Mathematics and Its Applications)
Xiaoping Xu
Manufacturer: Springer
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ASIN: 0792352424 |
Book Description
This book presents a systematic study on the structures of vertex operator superalgebras and their modules. Related theories of self-dual codes and lattices are included, as well as recent achievements on classifications of certain simple vertex operator superalgebras and their irreducible twisted modules, constructions of simple vertex operator superalgebras from graded associative algebras and their anti-involutions, self-dual codes and lattices.
Audience: This book is of interest to researchers and graduate students in mathematics and mathematical physics.
Book Description
This is the softcover reprint of the English translation of 1974 (available from Springer since 1989) of the first 3 chapters of Bourbaki's 'Algèbre'. It gives a thorough exposition of the fundamentals of general, linear and multilinear algebra. The first chapter introduces the basic objects: groups, actions, rings, fields. The second chapter studies the properties of modules and linear maps, especially with respect to the tensor product and duality constructions. The third chapter investigates algebras, in particular tensor algebras. Determinants, norms, traces and derivations are also studied.
Book Description
From reviews of the first edition: "The important feature of the present book is that it starts from the beginning (with only a very modest knowledge assumed) and covers all important topics... The book is very carefully organized [and] ends with 20 pages of useful historic comments. Such a comprehensive and carefully written treatment of fundamentals of the theory will certainly be a basic reference and text book in the future." -- Newsletter of the EMS "This is a fundamental book and none, beginner or expert, could afford to ignore it. Some results are really difficult to be found in other monographs, while others are for the first time included in a book." -- Mathematica "Each chapter begins with an excellent summary of the content and ends with an exercise section... This is really an outstanding book, well written and beautifully produced. It is both a graduate text and a monograph, so it can be recommended to graduate students as well as to specialists." -- Publicationes Mathematicae Lie Groups Beyond an Introduction takes the reader from the end of introductory Lie group theory to the threshold of infinite-dimensional group representations. Merging algebra and analysis throughout, the author uses Lie-theoretic methods to develop a beautiful theory having wide applications in mathematics and physics. A feature of the presentation is that it encourages the reader's comprehension of Lie group theory to evolve from beginner to expert: initial insights make use of actual matrices, while later insights come from such structural features as properties of root systems, or relationships among subgroups, or patterns among different subgroups. Topics include a description of all simply connected Lie groups in terms of semisimple Lie groups and semidirect products, the Cartan theory of complex semisimple Lie algebras, the Cartan-Weyl theory of the structure and representations of compact Lie groups and representations of complex semisimple Lie algebras, the classification of real semisimple Lie algebras, the structure theory of noncompact reductive Lie groups as it is now used in research, and integration on reductive groups. Many problems, tables, and bibliographical notes complete this comprehensive work, making the text suitable either for self-study or for courses in the second year of graduate study and beyond.
Customer Reviews:
Review of Knapp's "Lie groups: beyond an introduction.".......2002-08-13
The short version: this is a superbly written and conceived book; if I had to learn this material (the basic theory of
structure and representation of Lie algebras and groups,
especially semimsimple ones) from a single book, this is
the one I'd choose, among those I've seen. If you know the
basics of abstract algebra and some very basic concepts from
topology and manifolds, and you want to learn this material,
use this book. It would be a good reference, too, as it is
easy to find things in it, and takes a fairly modern, sophisticated approach (without sacrificing motivation and
intuition).
The long version, if you want more convincing or details:
I have used several books recently in learning the structure and
representation theory of Lie algebras and groups (especially Humphreys' Introduction to Lie algebras and representation theory, Fulton
and Harris' "Representation Theory," Varadarajan's "Lie groups,
Lie algebras, and their representations.") Although I came to Knapp's book with a decent background from the others, I think it's the best pedagogically, for someone with a modicum of mathematical sophistication and some basics like abstract
algebra and an idea of what a smooth manifold is), and a smattering of Lie theory. Some examples of the book's strength:
Elementary but potentially confusing concepts (like complexification, real forms, field extensions)
are explained thoroughly but in a sophisticated way, rather
than viewed as obvious. Carefully chosen examples motivate and
clarify the general theory; consequently even though the book
is completely rigorous, and carefully delineates lemmas, proofs,
remarks, definitions, and the like, it seems less dry then some
others (e.g. Varadarajan, from my point of view). But the point
of the examples, and their relation to the general theory, is
made clear, so they do not provide an overload of detail or b
obscure the main structure. Thought is always given to the
reader's understanding, not just to logical correctness, though
the author also takes the point of view, with which I concur,
that logical clarity and sufficient detail are essential
to understanding. Relations between ideas, alternative
proofs, and the structure of the theory to come are discussed
thoroughly, but such discussion is clearly demarcated from
the main structure of the argument, so that the latter is never
obscured. This is a fantastic book, and exactly what I was
looking for. Whether you are learning the material for the
first time, or want to review it or refer to, it is a superb
source.
Book Description
Clearly presented elements of one of the most penetrating concepts in modern mathematics include discussions of fields, vector spaces, homogeneous linear equations, extension fields, polynomials, algebraic elements, as well as sections on solvable groups, permutation groups, solution of equations by radicals, and other concepts. 1966 edition.
Customer Reviews:
the source!.......2004-04-13
This is modern Galois Theory, straight from the horse's mouth! Galois Theory is taught today using field extensions rather than by actually solving polynomials, students also learn to view a field extension as a vector space over the smaller field; both of these things were pioneered by Artin. The book also has short, clear proofs of all the main theorems. The only problem is that there are no problems to work on, so I have to say this is only a good reference for Galois Theory.
Succinct exposition of modern Galois theory by a pioneer........2003-11-13
Emil Artin's short book gets a mention in most texts on
Galois theory. It is very short - only 60 odd pages. Yet
it is a very clear, complete and readable account of the
essential elements of modern Galois theory. It is based
on lectures he gave over 50 years ago but you might think
it was written only yesterday and is comprehensible to
anyone familiar with current abstract algebra terminology.
And the price makes it a bargain. There are no worked
examples, exercises or index here.
just enjoy.......2002-02-19
during reading this cute booklet, you can surely hear the gentle talk of an old math maven.(from the publishing date, the auther was 44 but that's my impression.) with a cup of coffee, stretch those edgy wrinkles of your brain.
Nicely writien, short........2001-07-20
A friend of mine has a maxim: The shorter a math book, the more likely I am to read it. Artin's Galois Theory is certainly that. It is also an example of Artin's wonderful mathematical style. Gian-Carlo Rota, who took classes from Artin when he was at princeton, said that Artin's proofs were perfect, as though he had gone through all the available proofs to find *the* proof and that was the one he used. Rota felt that this left the student at a disadvantage in that he didn't know about the effort that went into the proof, nor why it is beautiful. I disagree: the proofs in Galois Theory have a certain indescribable beauty to them which left me awestruck at their simplicity. They seem to have all the requisite attributes (as laid out in Hardy's A Mathematician's apology) to be considered beautiful. These notes are by no means complete, but I would suggest them as a suplement to another treatment of field theory (for example, Dummit and Foote or Morandi even though they were based upon Artin's treatment).
Okay if you are interested in matehmatical "classics"........2001-05-14
I agree, to some extent, with the recent two reviewers: Nobody can deny that Emil Artin was a great mathematician, having done a very good job in algebra. That does not necessarily mean his textbooks should be praised *ad infinitum*. I understand some classics remain valuable for an incredibly long period of time ("Morse theory" by Milnor is one of such landmarks that comes into my mind), but I feel scheptical if this one deserves that claim. This book is okay if you are interested in his writing style of many years ago, but not quite so if your main concern is to study Galois theory (or algebra: that makes no difference for that matter) efficiently and effectively. In that case you should turn to more modern textbooks like Cohn ("Algebra" published by Wiley.)
Book Description
Lie groups has been an increasing area of focus and rich research since the middle of the 20th century. Procesi's masterful approach to Lie groups through invariants and representations gives the reader a comprehensive treatment of the classical groups along with an extensive introduction to a wide range of topics associated with Lie groups: symmetric functions, theory of algebraic forms, Lie algebras, tensor algebra and symmetry, semisimple Lie algebras, algebraic groups, group representations, invariants, Hilbert theory, and binary forms with fields ranging from pure algebra to functional analysis.
Key to this unique exposition is the large amount of background material presented so the book is accessible to a reader with relatively modest mathematical background. Historical information, examples, exercises are all woven into the text.
Lie Groups: An Approach through Invariants and Representations will engage a broad audience, including advanced undergraduates, graduates, mathematicians in a variety of areas from pure algebra to functional analysis and mathematical physics.
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Introductory Lectures on Rings and Modules (London Mathematical Society Student Texts)
John A. Beachy
Manufacturer: Cambridge University Press
ProductGroup: Book
Binding: Paperback
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ASIN: 0521644070 |
Book Description
The focus of this book is the study of the noncommutative aspects of rings and modules, and the style will make it accessible to anyone with a background in basic abstract algebra. Features of interest include an early introduction of projective and injective modules; a module theoretic approach to the Jacobson radical and the Artin-Wedderburn theorem; the use of Baer's criterion for injectivity to prove the structure theorem for finitely generated modules over a principal ideal domain; and applications of the general theory to the representation theory of finite groups. Optional material includes a section on modules over the Weyl algebras and a section on Goldie's theorem. When compared to other more encyclopedic texts, the sharp focus of this book accommodates students meeting this material for the first time. It can be used as a first-year graduate text or as a reference for advanced undergraduates.
Customer Reviews:
Clear and concise........2005-01-25
This is a nice little book. It's pretty thorough despite its size, and though dense it is clear enough to be easy to follow.
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