This book's organizing principle is the interplay between groups and rings, where "rings" includes the ideas of modules. It contains basic denitions, complete and clear theorems (the rst with brief sketches of proofs), and gives attention to the topics of algebraic geometry, computers, homology, and representations. More than merely a succession of denition-theorem-proofs, this text put results and ideas in context so that students can appreciate why a certain topic is being studied, and where denitions originate. Advanced Algebra systematically develops concepts and tools in algebra that are vital to every mathematician, whether pure or applied, aspiring or established. The book aims to give the reader a global view of algebra, its use, and its role in mathematics as a whole. The idea is to explain what the young mathematician needs to know about algebra in order to communicate well with colleagues in all branches of mathematics. The book is written as textbooks, and the primary audience is students who are learning the material for the rst time and who are planning a career in which they will use advanced mathematics professionally. Much of the material in the book, including nearly all of Basic Algebra and some of Advanced Algebra, corresponds to normal course work, with the proportions depending on the university. The books include further topics that may be skipped in required courses but that the professional mathematician will ultimately want to learn by self-study. This book can be used as a text for the rst year of graduate algebra, but it is much more than that. It can also serve more advanced graduate students wishing to learn topics on their own; while not reaching the frontiers, the book does provide a sense of the successes and methods arising in an area. Finally, this is a reference containing many of the standard theorems and denitions that users of algebra need to know. Thus, the book is not only an appetizer, but a hearty meal as well.
Professor Cyrus Hettle School of Mathematics Georgia Institute of Technology Atlanta, is a mathematician known for his work in areas like abstract algebra, combinatorics, and the history of mathematics, particularly his research on Diophantus. He has explored the symbolic and mathematical inuence of Diophantus's Arithmetica, bridging the gap between ancient and modern mathematical notation. His mathematical research lies at the interface between number theory, algebraic geometry, complex dynamics, and combinatorics. He completed PhD in Mathematics in December 2000. Thesis: Computational Complexity of Generalized Pattern Matching. Advisor: John Rhodes from University of California at Berkeley. He holds, BS with Highest Distinction in Mathematics, Magna Cum Laude, May 1994 from University of Illinois at Urbana-Champaign.
Preface............................................................................................................ v Chapter 1. Fundamental Algebraic Structures..................................................... 1 Groups: Definition and Examples.............................................................. 2 Subgroups and Cosets..............................................................................10 Normal Subgroups and Quotient Groups................................................15 Group Homomorphisms and Isomorphisms............................................19 Chapter 2. Advanced Group Theory.................................................................. 26 Cyclic Groups and Generators.................................................................. 27 Permutation Groups and Symmetric Groups.......................................... 32 Group Actions and Orbit-Stabilizer Theorem..........................................39 Sylow Theorems.......................................................................................46 Chapter 3. Ring Theory....................................................................................... 49 Integral Domains and Zero Divisors........................................................50 Ideals and Quotient Rings........................................................................ 53 Prime and Maximal Ideals........................................................................ 57 Ring Homomorphisms and Isomorphisms..............................................63 Chapter 4. Field Theory and Galois Theory....................................................... 67 Algebraic and Transcendental Extensions..............................................68 Splitting Fields and Normal Extensions................................................... 73 Separable and Inseparable Extensions................................................... 75 The Fundamental Theorem of Galois Theory.......................................... 81 Chapter 5. Modules and Vector Spaces............................................................. 84 Modules over a Ring: Definitions and Examples....................................85 Submodules and Quotient Modules........................................................88 Linear Independence and Bases of Vector Spaces.................................92 Dimension Theorem and Rank-Nullity.....................................................96 Chapter 6. Advanced Linear Algebra................................................................. 99 Inner Product Spaces and Orthogonality.............................................. 100 Contents Diagonalization and Jordan Canonical Form........................................ 106 Bilinear and Quadratic Forms................................................................. 110 Spectral Theorem for Normal Operators............................................... 117 Chapter 7. Commutative
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