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A comprehensive presentation of abstract algebra and an in-depth treatment of the applications of algebraic techniques and the relationship of algebra to other disciplines, such as number theory, combinatorics, geometry, topology, differential equations, and Markov chains. ... Read more

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Abstract Algebra vol 2 by Spindler
Page 174 volume 2 the author did not follow completely the proof of Nagell theroem as in Algebraic Number Theory by Stewart & TallInstead Heintroduced the matrix [1 1, -2 0], We may name it here for short S, Theauthor claims that S**42k = [-1 -1, -1 -1] mod 3 which is wrong S**42k =+-[1 0, 1 0] 4|k or 2|k respectively and = +-S 4|k-1 or 2|k-1 respectivelyin both cases the first entery in the total matrix as S**(42k +13) * [0 1]would be -1 and destroy the logic of the author in his argument However itdeos not violate the theory for -1 in a field of chratersistic 3 need notbe -1 in the integral domain Z[sqrt(-7)] I do here thank Amazon.com for thechance to report that for neithe the author nor the editorial at thepublisher has an address to communicate with. Abdu Shokr35-17 Ditmarsblvd suite 202 NY NY 11105 August 3 2000

A thorough, accessible and indispensable resource on algebra
This two volume set is by far the most complete text on abstract algebra available.Rich in detail, a wealth of examples and clear yet concise explanations provide the reader an accesible source to deepen his understanding of abstract algebra.Spindler conveys his love of algebraand a desire to reveal its riches to the average student. If you'verelied on Hungerford's / Lang's Algebra or Warner's Modern Algebra, thistext is the last text you will need. ... Read more

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The purpose of this book is to prepare the reader for coping with abstract mathematics. The intended audience is both students taking a first course in abstract algebra who feel the need to strengthen their background, and those from a more applied background who need some experience in dealing with abstract ideas. Learning any area of abstract mathematics requires not only ability to write formally but also to think intuitively about what is going on and to describe that process clearly and cogently in ordinary English. Ash tries to aid intuition by keeping proofs short and as informal as possible, and using concrete examples as illustration. Thus it is an ideal textbook for an audience with limited experience in formalism and abstraction. A number of expository innovations are included, for example, an informal development of set theory which teaches students all the basic results for algebra in one chapter. ... Read more

Customer Reviews (3)

Concise and Direct
I have used many of Robert Ash's book, including "Information Theory", which was his first published textbook based on notes he used at Columbia University to teach a course by the same name and which were subsequently used by other instructors, and "Basic Probability Theory".His books all have the same characteristic: simple direct introduction to the subject, powerful penetration to the core of the subject matter with no clutter and a deceptively comprehensive coverage in a succint manner. Any time spent with any of his books is well spent.In this instance I can not think of a better introduction to abstract mathematics. As another reviewer noted, the treatment of linear algebra is evidence of Ash's concise and powerful method. Having said this I would strongly reccomend, as Ash himself does, that the reader consult other books to flesh out the treatment, or better still to create and work through his own examples and exercises. Highly recommended!

A good primer of mathematics
I would alter the title by dropping the word abstract, as the book could be considered a primer on the mathematics of an undergraduate major. I recently completed teaching a course in fundamentals of logic and arithmetic for elementary education majors and many of the topics we covered are in here. The basics of logic, counting, sets and functions, the content of the first two chapters of this book, were all in the fundamentals class, although obviously not to this level of detail. Factoring, unique factorization, modular arithmetic, and the cardinality of sets, covered in chapters four and five, were also the subjects of several of my lectures. Only the principles of linear algebra and linear operators, the topics of chapters 5 and 6, are in this book but were not in my class.
The coverage here is basic, with a set of problems at the end of each section. Solutions to all of the problems are given at the end of the book, a very commendable thing to have done. All of the topics covered are explained in a routine, yet thorough manner. By that I mean that the explanatory strategy is largely a traditional one, with definitions, theorems, proofs and interspersed examples where appropriate. Diagrams are used, but only rarely.
Solid, but not spectacular, this book covers the fundamentals of mathematics that all students should know before they begin calculus and linear algebra. It can serve as a primer or a refresher, depending on whether you are moving forward or are retracing old ground.

Published in the recreational mathematics e-mail newsletter, reprinted with permission.

A good primer of mathematics
I would alter the title by dropping the word abstract, as the book could be considered a primer on the mathematics of an undergraduate major. I recently completed teaching a course in fundamentals of logic and arithmetic for elementary education majors and many of the topics we covered are in here. The basics of logic, counting, sets and functions, the content of the first two chapters of this book, were all in the fundamentals class, although obviously not to this level of detail. Factoring, unique factorization, modular arithmetic, and the cardinality of sets, covered in chapters four and five, were also the subjects of several of my lectures. Only the principles of linear algebra and linear operators, the topics of chapters 5 and 6, are in this book but were not in my class.
The coverage here is basic, with a set of problems at the end of each section. Solutions to all of the problems are given at the end of the book, a very commendable thing to have done. All of the topics covered are explained in a routine, yet thorough manner. By that I mean that the explanatory strategy is largely a traditional one, with definitions, theorems, proofs and interspersed examples where appropriate. Diagrams are used, but only rarely.
Solid, but not spectacular, this book covers the fundamentals of mathematics that all students should know before they begin calculus and linear algebra. It can serve as a primer or a refresher, depending on whether you are moving forward or are retracing old ground.

Published in the recreational mathematics e-mail newsletter, reprinted with permission. ... Read more

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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

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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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This undergraduate text takes a novel approach to the standard introductory material on groups, rings, and fields. At the heart of the text is a semi-historical journey through the early decades of the subject as it emerged in the revolutionary work of Euler, Lagrange, Gauss, and Galois. Avoiding excessive abstraction whenever possible, the text focuses on the central problem of studying the solutions of polynomial equations. Highlights include a proof of the Fundamental Theorem of Algebra, essentially due to Euler, and a proof of the constructability of the regular 17-gon, in the manner of Gauss. Another novel feature is the introduction of groups through a meditation on the meaning of congruence in the work of Euclid. Everywhere in the text, the goal is to make clear the links connecting abstract algebra to Euclidean geometry, high school algebra, and trigonometry, in the hope that students pursuing a career as secondary mathematics educators will carry away a deeper and richer understanding of the high school mathematics curriculum. Another goal is to encourage students, insofar as possible in a textbook format, to build the course for themselves, with exercises integrally embedded in the text of each chapter. ... Read more

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worst textbook ever!
ridiculously expensive and unbelievable thin with not much materials inside, the worst textbook I've ever had in my life! ... Read more

Customer Reviews (1)

a great book for non-math majors
Abstract Algebra, A Computational Approach is an introductory textbook on abstract algebra that emphasises the "hands on" approach of computing and calculating everything imaginable: Sets, groups, rings,fields, and the operations on them.

This wonderful book struggles with amajor problem in math education: How do you get a class of 40 students tovisualise abstract algebra? Ideally, you'd want to give many examples andproblems for them to work out. Soviet books have traditionally taken thisapproach, so with many Soviet math books there was a compendium of"1000 solved problems in... ". There are similar books inEnglish, but only for the most elementary courses. Having a few dosen bookswith solved problems in Topology, for example, would have been a greatthing, but there isn't any economic motivation for such books. Alternately,if it were possible to calculate and compute things, then students couldexperiment with and explore the subject matter on their own. This is whatgraphing calculators, for example, did to intro calculus classes -- it madeit possible for students to experiment and visualise the subject matterlike nothing else.

But how do you visualise abstract algebra? You couldwrite computer programs to do such things as compute permutations,intersect sets, solve matrices, etc, but in "mainstream"programming languages this would take a while, be error prone and slow downthe class. It wouldn't be practical. The author's solution: Use APL! APL isa programming language designed specifically for concisely manipulatingvectors and matrices. APL programs are so concise that in one or two linesof APL code students can accomplish what would otherwise take hundreds oflines of C code. With a few keystrokes, students could be performingcomplex operations on groups and fields and sets. APL is fun, quick andconcise, and students will be focused on what really matters -- abstractalgebra, rather than delving into the nuts and bolts o yet-anotherprogramming language. APL to abstract algebra is like a graphing calculatorto studying functions -- it helps, it's fun, it gives intuition andinsight.

The book doesn't cover as much material as, say, Herstein'sbook. On the other hand, it will give the students a quick feel for thesubject matter. Perhaps the book would be ideal for computer science majorsand other non-math majors that are expected to take two semesters ofabstract algebra. Another minor issue is that the dialect of APL used inthe book is a bit old -- the code's great, but I think the students wouldhave a better experience with APL2 or J. J has a richer numeric tower andmuch of the supporting software is made redundant by it. ... Read more

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Praise for the first edition

"This book is clearly written and presents a large number of examples illustrating the theory . . . there is no other book of comparable content available. Because of its detailed coverage of applications generallyneglected in the literature, it is a desirable if not essential addition to undergraduate mathematics and computer science libraries."
–CHOICE

As a cornerstone of mathematical science, the importance of modern algebra and discrete structures to many areas of science and technology is apparent and growing–with extensive use in computing science, physics, chemistry, and data communications as well as in areas of mathematics such as combinatorics.

Blending the theoretical with the practical in the instruction of modern algebra, Modern Algebra with Applications, Second Edition provides interesting and important applications of this subject–effectively holding your interest and creating a more seamless method of instruction.

Incorporating the applications of modern algebra throughout its authoritative treatment of the subject, this book covers the full complement of group, ring, and field theory typically contained in a standard modern algebra course. Numerous examples are included in each chapter, and answers to odd-numbered exercises are appended in the back of the text.

Chapter topics include:

• Boolean Algebras
• Polynomial and Euclidean Rings
• Groups
• Quotient Rings
• Quotient Groups
• Field Extensions
• Symmetry Groups in Three Dimensions
• Latin Squares
• Pólya—Burnside Method of Enumeration
• Geometrical Constructions
• Monoids and Machines
• Error-Correcting Codes
• Rings and Fields

In addition to improvements in exposition, this fully updated Second Edition also contains new material on order of an element and cyclic groups, more details about the lattice of divisors of an integer, and new historical notes.

Filled with in-depth insights and over 600 exercises of varying difficulty, Modern Algebra with Applications, Second Edition can help anyone appreciate and understand this subject. ... Read more

Customer Reviews (3)

Good supplemental book
I just bought this title after having the second edition checked out from university library most of the semester. Be careful not to confuse the two: Amazon's system didn't make it clear that this was the edition being sold. Both are fine books and the first ed is a steal at the price it is commonly offered at, but the the second ed is quite a bit more concise with a second author contributing to the updated materials.This version seems to have considerably more material dedicated to applications, and would be a good additional reference to have on hand whether for introductory abstract algebra at the undergrad level or grad, or an applications course.

The best modern algebra text I have seen
It has been some time since I taught a course in modern algebra, but if I were to teach it again, I would use this book. The first chapter covers the basics of binary operations and algebraic structures. Seven pages in length, it is a basic overview of what the students should already know. Chapter two covers what is arguably the simplest abstract algebra, Boolean algebra. Propositional logic, switching circuits, posets and lattices are the examples used. I am in strong agreement with the introduction of switching circuits this early in a course. It is an application of abstract algebra that students can easily understand and relate to. Switching circuits also immediately demonstrates that the inclusion of "with applications" in the title is not a misnomer.
Basic group theory is the topic of chapter three, quotient and symmetry groups are covered in chapters four and five. The Polya-Burnside method of enumeration is the topic of chapter five, monoids and machines the focus of chapter seven, rings, fields, polynomial, Euclidean and quotient rings are examined in chapters eight, nine and ten. Field extensions are covered in chapter eleven, Latin squares in chapter twelve, geometrical constructions in chapter thirteen and error-correcting codes in chapter fourteen.
The exposition is excellent, it is one of the more readable texts in advanced mathematics that you will find. There are many examples of the applications of modern algebra to the real world and a large number of exercises are included at the end of the chapters. Solutions to the odd numbered exercises are found in an appendix.
Modern algebra is a difficult topic due to the abstract nature of the material. It is easy for students to get confused and lost when they are studying the book on their own. The risk of that is minimized if they are reading this book.

important computer applications
This book is pitched at the level of a third or fourth undergraduate year in maths, or at a graduate level course. Gilbert assumes a solid background in maths, especially linear algebra. He expounds on groups and rings.

You can certainly treat this book as a pure maths text on algebra. But there is a twist to it, for some of you. He discusses applications in engineering. Notably Boolean algebra. Boolean logic forms the conceptual and practical basis of computer hardware and software. Gilbert shows how to represent a Boolean expression in a normal form and how to simplify it. Very practical and vital to circuit design.

Later on in the book, he gives a good but all too brief discussion of finite state machines. These can be used in control systems and even in the construction of computer languages like Java. Where the Java bytecode can be validated precisely because Java is a finite state machine.

Finally, error correcting codes take up an entire chapter. Important in communications theory, and even in such common items as storing information on computer media like DVDs. ... Read more

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This is a revised and expanded edition of Kac's original introduction to algebraic aspects of conformal field theory, which was published by the AMS in 1996. The volume serves as an introduction to algebraic aspects of conformal field theory, which in the past 15 years revealed a variety of unusual mathematical notions. Vertex algebra theory provides an effective tool to study them in a unified way. In the book, a mathematician encounters new algebraic structures that originated from Einstein's special relativity postulate and Heisenberg's uncertainty principle. A physicist will find familiar notions presented in a more rigorous and systematic way, possibly leading to a better understanding of foundations of quantum physics. This revised edition is based on courses given by the author at MIT and at Rome University in spring 1997. New material is added, including the foundations of a rapidly growing area of algebraic conformal theory. Also, in some places the exposition has been significantly simplified. ... Read more

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This is the most current textbook in teaching the basic concepts of abstract algebra. The author finds that there are many students who just memorise a theorem without having the ability to apply it to a given problem. Therefore, this is a hands-on manual, where many typical algebraic problems are provided for students to be able to apply the theorems and to actually practice the methods they have learned. Each chapter begins with a statement of a major result in Group and Ring Theory, followed by problems and solutions. ... Read more

Customer Reviews (3)

its great
actually i liked it because it has strong proofs. i think the other people did not like it because they think it is a solution for an introduction course ITS NOT u should know the subject first.

Not worth the paper it is printed on
There are better books available. If you blindly pick up a book, I am sure that it will be better than this one.

my opinion
Just as long as a book has all thesolutionstoits problems, it will always bea good book.Otherwise, it is just plain talk. ... Read more

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Based on years of experience teaching and writing supplemental materials for more traditional precalculus texts, Reva Narasimhan takes a functions-focused approach to teaching and learning algebra and trigonometry concepts. This new series builds up relevant concepts using functions as a unifying theme, repeating and expanding on connections to basic functions.Visualization and analysis motivate the functions-based approach, enabling students to better retain the material for use in later calculus courses. ... Read more

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Evaluation of Abstract Algebra book purchase.
The book was essentially new.No marks, and it was exactly what I was looking for. ... Read more

Customer Reviews (2)

Exact unified language for physics and mathematics
i have been working a few years in geometric calculus and i believe this book should be in every house of every geometrist and every person that is intersted in geometric concepts with physics applications

a marriage of geometry and algebra made in heaven
In 1990, I came across Hestenes' book New Foundations for Classical Mechanics in a university bookstore, and immediately purchased it. In 1999, I finally purchased the paperback version of CA to GC (Kluwer, please keep the paperback in print!).An engineer by training, I was previously only familiar with college vector analysis and with indicial tensor notation.These two books revolutionized my understanding of the algebraic representation of geometric information.Amazon.com now has the "look inside" feature in place for this book, so you can check the table of contents.

Science proceeds both by discovery and by a process of recasting what has been learned in simpler and clearer form.It is the fruits of this latter process that are presented in CA to GC, though some new mathematical results are also introduced.The authors have succeeded admirably in recasting large areas of intermediate to advanced mathematics in a powerful unified algebraic language of exceptional clarity.The authors show how the traditional languages of complex numbers, quaternions, matrices, vectors, tensors, spinors and differential forms are all subsumed by the elegant language of Clifford algebra, and their calculi by Clifford analysis.Quite apart from the pleasure that the clarity of Clifford algebra/analysis affords, its value also lies in making it easier to understand what has already been discovered, and thus extending the mathematical grasp of the human mind.I have often seen the terms "breakthrough" and "groundbreaking" applied to paltry advances in science, mostly by the innovators themselves, but surely Clifford algebra/analysis is deserving of such an appellation.It has been long in gestation, but its time has come.

In CA to GC, the authors present a tour-de-force of mathematical exposition, the writing displaying the same perspicuity and precision that marks all of Hestenes' writing.While further-refined versions of much of the material of the book can now be downloaded in the form of pdf files from Hestenes' website, this book will go down in history as a classic of unifying mathematical exposition.The university student should begin with the New Foundations for Classical Mechanics book, but CA to GC should be read by every mathematician, physicist and engineering scientist.The reader contemplating learning Clifford algebra/analysis should also take a look at the rapidly growing amount of information online, and at other books on the subject.It is an honor for me to be the first reviewer of this book on Amazon.com. ... Read more

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Practice Makes Perfect has established itself as a reliable practical workbook series in the language-learning category. Practice Makes Perfect: Algebra, provides students with the same clear, concise approach and extensive exercises to key fields they've come to expect from the series-but now within mathematics. This book presents thorough coverage of skills, such as handling decimals and fractions, functions, and linear and quadratic equations.
Practice Makes Perfect: Algebra is not focused on any particular test or exam, but complementary to most algebra curricula. Because of this approach, the book can be used by struggling students needing extra help, readers who need to firm up skills for an exam, or those who are returning to the subject years after they first studied algebra.