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PART I: INTEGERS AND EQUIVALENCE RELATIONS. Preliminaries. Properties of Integers. Complex Numbers. Modular Arithmetic. Mathematical Induction. Equivalence Relations. Functions (Mappings). Exercises. PART I: GROUPS. 1. Introduction to Groups. Symmetries of a Square. The Dihedral Groups. Exercises. Biography of Neils Abel 2. Groups. Definition and Examples of Groups. Elementary Properties of Groups. Historical Note. Exercises. 3. Finite Groups; Subgroups. Terminology and Notation. Subgroup Tests. Examples of Subgroups. Exercises. 4. Cyclic Groups. Properties of Cyclic Groups. Classification of Subgroups of Cyclic Groups. Exercises. Biography of J. J. Sylvester. Supplementary Exercises for Chapters 1-4. 5. Permutation Groups. Definition and Notation. Cycle Notation. Properties of Permutations. A Check-Digit Scheme Based on D5. Exercises. Biography of Augustin Cauchy. 6. Isomorphisms. Motivation. Definition and Examples. Cayley's Theorem. Properties of Isomorphisms. Automorphisms. Exercises. Biography of Arthur Cayley. 7. Cosets and Lagrange's Theorem. Properties of Cosets. Lagrange's Theorem and Consequences. An Application of Cosets to Permutation Groups. The Rotation Group of a Cube and a Soccer Ball. Exercises. Biography of Joseph Lagrange. 8. External Direct Products. Definition and Examples. Properties of External Direct Products. The Group of Units Modulo n as an External Direct Product. Applications. Exercises. Biography of Leonard Adleman. Supplementary Exercises for Chapters 5-8 9. Normal Subgroups and Factor Groups. Normal Subgroups. Factor Groups. Applications of Factor Groups. Internal Direct Products. Exercises. Biography of Evariste Galois 10. Group Homomorphisms. Definition and Examples. Properties of Homomorphisms. The First Isomorphism Theorem. Exercises. Biography of Camille Jordan. 11. Fundamental Theorem of Finite Abelian Groups. The Fundamental Theorem. The Isomorphism Classes of Abelian Groups. Proof of the Fundamental Theorem. Exercises. Supplementary Exercises for Chapters 9-11. PART III: RINGS. 12. Introduction to Rings. Motivation and Definition. Examples of Rings. Properties of Rings. Subrings. Exercises. Biography of I. N. Herstein. 13. Integral Domains. Definition and Examples. Fields. Characteristic of a Ring. Exercises. Biography of Nathan Jacobson. 14. Ideals and Factor Rings. Ideals. Factor Rings. Prime Ideals and Maximal Ideals. Exercises. Biography of Richard Dedekind. Biography of Emmy Noether. Supplementary Exercises for Chapters 12-14. 15. Ring Homomorphisms. Definition and Examples. Properties of Ring Homomorphisms. The Field of Quotients. Exercises. 16. Polynomial Rings. Notation and Terminology. The Division Algorithm and Consequences. Exercises. Biography of Saunders Mac Lane. 17. Factorization of Polynomials. Reducibility Tests. Irreducibility Tests. Unique Factorization in Z[x]. Weird Dice: An Application of Unique Factorization. Exercises. Biography of Serge Lang. 18. Divisibility in Integral Domains. Irreducibles, Primes. Historical Discussion of Fermat's Last Theorem. Unique Factorization Domains. Euclidean Domains. Exercises. Biography of Sophie Germain. Biography of Andrew Wiles. Supplementary Exercises for Chapters 15-18. PART IV: FIELDS. 19. Vector Spaces. Definition and Examples. Subspaces. Linear Independence. Exercises. Biography of Emil Artin. Biography of Olga Taussky-Todd. 20. Extension Fields. The Fundamental Theorem of Field Theory. Splitting Fields. Zeros of an Irreducible Polynomial. Exercises. Biography of Leopold Kronecker. 21. Algebraic Extensions. Characterization of Extensions. Finite Extensions. Properties of Algebraic Extensions Exercises. Biography of Irving Kaplansky. 22. Finite Fields. Classification of Finite Fields. Structure of Finite Fields. Subfields of a Finite Field. Exercises. Biography of L. E. Dickson. 23. Geometric Constructions. Historical Discussion of Geometric Constructions. Constructible Numbers. Angle-Trisectors and Circle-Squarers. Exercises. Supplementary Exercises for Chapters 19-23. PART V: SPECIAL TOPICS. 24. Sylow Theorems. Conjugacy Classes. The Class Equation. The Probability That Two Elements Commute. The Sylow Theorems. Applications of Sylow Theorems. Exercises. Biography of Ludvig Sylow. 25. Finite Simple Groups. Historical Background. Nonsimplicity Tests. The Simplicity of A5. The Fields Medal. The Cole Prize. Exercises. Biography of Michael Aschbacher. Biography of Daniel Gorenstein. Biography of John Thompson. 26. Generators and Relations. Motivation. Definitions and Notation. Free Group. Generators and Relations. Classification of Groups of Order up to 15. Characterization of Dihedral Groups. Realizing the Dihedral Groups with Mirrors. Exercises. Biography of Marshall Hall, Jr. 27. Symmetry Groups. Isometries. Classification of Finite Plane Symmetry Groups. Classification of Finite Group of Rotations in R^3. Exercises. 28. Frieze Groups and Crystallographic Groups. The Frieze Groups. The Crystallographic Groups. Identification of Plane Periodic Patterns. Exercises. Biography of M. C. Escher. Biography of George Polya. Biography of John H. Conway. 29. Symmetry and Counting. Motivation. Burnside's Theorem. Applications. Group Action. Exercises. Biography of William Burnside. 30. Cayley Digraphs of Groups. Motivation. The Cayley Digraph of a Group. Hamiltonian Circuits and Paths. Some Applications. Exercises. Biography of William-Rowan Hamilton. Biography of Paul Erdos. 31. Introduction to Algebraic Coding Theory. Motivation. Linear Codes. Parity-Check Matrix Decoding. Coset Decoding. Historical Note: The Ubiquitous Reed-Solomon Codes. Exercises. Biography of Richard W. Hamming. Biography of Jessie MacWilliams. Biography of Vera Pless. 32. An Introduction to Galois Theory. Fundamental Theorem of Galois Theory. Solvability of Polynomials by. Radicals. Insolvability of a Quintic. Exercises. Biography of Philip Hall. 33. Cyclotomic Extensions. Motivation. Cyclotomic Polynomials. The Constructible Regular n-gons. Exercises. Biography of Carl Friedrich Gauss. Biography of Manjul Bhargava. Supplementary Exercises for Chapters 24-33. --This text refers to the Paperback edition.

About the Author

Joseph Gallian earned his PhD from Notre Dame. In addition to receiving numerous awards for his teaching and exposition, he served, first, as the Second Vice President, and, then, as the President of the MAA. He has served on 40 national committees, chairing ten of them. He has published over 100 articles and authored six books. Numerous articles about his work have appeared in the national news outlets, including the New York Times, the Washington Post, the Boston Globe, and Newsweek, among many others.

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Amazon.com: 12 reviews
22 of 22 people found the following review helpful
Best algebra book even if too expensive Sept. 22 2012
By C. Collomb - Published on Amazon.com
Format: Hardcover
I have been looking for a book to review some algebra I studied over 20 years ago, and after comparing hundreds of books (yes literally!) I am glad I finally discovered this one.

Why is this book so great?

1. It covers abstract algebra from basics to Galois theory including solvability by radicals, so you should not need any other book to cover the basics before Masters level
2. The book does not assume any prerequisite and is easy to follow even if you did not touch abstract Algebra for a while
3. The book is full of examples, basic groups had like over 20 of them making sure you really get them
4. All theorems have a proof, and the proofs are well explained
5. The material is split in over 30 chapters in nearly 600 pages, making each chapter the right size to make progress and not too long to discourage others
6. Each chapter has around 50/60 exercises with the solution for half of them, making it a practical training book
7. The book size is like 3 ipads stacked together, making it practical to take anywhere at the contrary of other references like the calculus books from salas and hille

Anyways, I am a big fan of this book. The price is really crazy though, so I recommend you get a used copy of the 5th edition instead. The content is almost identical, and in absolute mathematical value identical. Nothing is missing, just some different exercises and some different examples. And for around $30 there is nothing better you can get.
9 of 9 people found the following review helpful
Great Book, but Phenomenally Overpriced Aug. 30 2012
By Baze - Published on Amazon.com
Format: Hardcover
Gallian's clear, easy-to-read text will be an excellent guide to anyone who decides to take a course in undergraduate abstract algebra. Gallian writes only the bare necessities for the student to read. Initially, this simplicity may reflect a writer who doesn't care about the student learning too much. Nothing could be further from the truth- by keeping his paragraphs, descriptions, and proofs concise, Gallian makes it easier for the student to focus on the theorems taught throughout the book. The problems that Gallian assigns as the end of each chapter begin easily, and then progress to very challenging problems. This conveniently allows the student to do extra problems depending on their comfort level with the material.

Despite that the text is very well written, and that the problem sets are quite reasonable, I might dock off two stars- because of the price. Personally, I think that it's outrageous a used copy has to sell for nearly $200. I have a copy of the 7th Ed as well, and beside the problem sets being slightly different, the material is quite similar between the 7th and 8th editions. The 8th Ed, in effect, is useless. If I were you, I'd try to save my hard-earned pennies and purchase some other edition of this book.
22 of 26 people found the following review helpful
Decent Book... but... Sept. 1 2013
By Chris A. Christopherson - Published on Amazon.com
Format: Hardcover
I am a math teacher (teaching lower-level math at a university) who is reviewing abstract algebra after 20 years of not thinking much about it. I have amassed quite a number of abstract algebra books over the past few months and have been jumping around between them as I work through this stuff. I have decided to review several of them. So here's what I think of this one:

Gallian does do an admirable job of explaining concepts. There are decent examples. I think the book is a bit heavy on the applications especially near the beginning. Gallian does tend to be a bit wordy at times, but not bad. As I said, he does explain things pretty well. Based only on the actual text, I'd have given the book 4 stars.

So why only 3 stars? Because I gave the exercises only 2 stars. They very nearly enrage me. Okay, it's not so much that the exercises themselves are horrible. My gripe is that they tend to be either routine problems (that's not my gripe) or overly difficult proofs/sophisticated conceptual problems. There's not a lot in between. That's my first criticism. Next, Gallian seems to have made very little attempt to grade the problems in increasing difficulty. You look at number 3 and you're thinking about it for a week, until finally you are tired and just move on to the next section... when all along, the "meat and potatoes" problems are buried in the sorts of problems that tend to take many hours of pondering. In my opinion, that's not a good way to present the problems. Next, there needs to be MANY, many more routine, concept-building problems. The problems that are there are okay, they just need to be tripled in number with the sorts of routine concept-builders that aid in understanding and cementing definitions and methodology. Leave the abstract proofs for the end of the problem sets.

Also, and this is more of a philosophy issue, just put the damn solutions in the back of the book. It's not a secret (or at least it shouldn't be). As an instructor, I have no problem coming up with testing material on my own and I've never picked even problems from the books. In fact, I've never assigned a problem that my students haven't had the answer waiting when they need to confirm their performance. A student not knowing if he/she is doing things correctly is a sure recipe for reinforcement of bad mathematical habits when they are not doing things correctly. They need to see their faults RIGHT AWAY and clear them up. It's stupid to have a student spending weeks not knowing if he/she has the material down and going into a test that way. Immediate feedback is critical, in my opinion. And it's my opinion that all textbooks should have the solutions available. Solve and confirm. Solve and confirm. Solve and confirm. That's what builds understanding, not a mysterious cloud that a student gets lost within. Just because a math genius can rip through a problem set with no feedback, confirmation, or hint, it's no sign everyone should have to do that. Not everyone can paint a Mona Lisa and not everyone can internalize this stuff on first glance. This "mathematical maturity" argument is silliness. One does not attain "mathematical maturity" via secrecy, reinventing the wheel, or discovering all of mathematics on his/her own. And all the blood, sweat and tears spent working a problem or doing a proof is 100% in vain when the student does not know if he/she has done the thing correctly.

Anyway, getting off my soapbox, this book is decent and you could do much worse. But, at the same time, the book could be so much better with just a bit of refinement. So... not bad, but there are others I'm more fond of (which I will review).
1 of 1 people found the following review helpful
The proofs are so vague. If you are a ... Dec 17 2014
By Adam - Published on Amazon.com
Format: Hardcover Verified Purchase
The proofs are so vague. If you are a detail oriented mathematician, you will struggle with how little detail is given to almost every proof in this book.
2 of 3 people found the following review helpful
Great Algebraic Reference May 27 2013
By Jennifer Long - Published on Amazon.com
Format: Hardcover Verified Purchase
We used this book as we studied Group Theory. This book was incredibly helpful. I love all of the detailed proofs, helpful hints on problems, as well as interesting summaries of famous mathematicians related to this field. It is an accessible book, and worth the money.