Basic Algebra: (Cornerstones)

Aspiring and established mathematicians alike will benefit from this comprehensive treatment of algebra, which systematically develops the concepts and tools of the subject and simultaneously emphasizes a global view of the discipline. "Cornerstones of Algebra" stresses the fundamentals and highlights the connections between algebra and other branches of mathematics, including analysis, topology, and number theory. It requires of the reader only familiarity with some linear algebra and group theory, understanding of equivalence relations, basic knowledge of complex numbers, and an acquaintance with proofs. Key topics and features: * Chapters on linear algebra develop the notion of vector spaces, the theory of linear transformations, bilinear forms, classical linear groups, and multilinear algebra * Chapters on modern algebra treat groups, rings, fields, modules, and Galois theory, including infinite Galois groups * Later chapters cover more advanced topics in commutative and noncommutative algebra and provide introductions to algebraic number theory, algebraic geometry, and homological algebra * Two prominent themes recur throughout: the interplay between groups and linear algebra, and the relationships between number theory and geometry * Each chapter includes hundreds of examples and problems, and a final chapter contains hints and solutions for most of the problems Because it focuses on what every student needs to know about algebra, this book is ideal as a course text and for self-study, especially for graduate students preparing to take examinations. Its scope and approach will also appeal to professors in diverse areas. Indeed, the clarity and breadth of "Cornerstones of Algebra" make it a welcome addition to every mathematician's library.

Table of Contents:
Preliminaries about the Integers, Polynomials, and Matrices.- Vector Spaces over ?, ?, and ?.- Inner-Product Spaces.- Groups and Group Actions.- Theory of a Single Linear Transformation.- Multilinear Algebra.- Advanced Group Theory.- Commutative Rings and Their Modules.- Fields and Galois Theory.- Modules over Noncommutative Rings.

Review :
From the reviews: Finally, the author's notorious masterly style of writing, which stands out by its high degree of clarity, elegance, refinement, and accuracy, also rules over this newest textbook of his, which is very likely to become one of the great standard texts in algebra for generations. -Zentralblatt Math "Anthony W. Knapp Basic Algebra aims to teach the student algebra in a somewhat different way than is the norm. Together with its sequel, Advanced Algebra, the book proposes to take an aspiring mathematician from his first exposure to algebra proper, right after calculus, through his graduate training in algebra -- and a bit further... "To use Basic Algebra in a standard undergraduate curriculum would entail spreading the material over several semesters, since the book can be used for linear algebra, group theory (Chapter IV is a gem!), rings and fields, Galois theory, a second course in group theory, and even a course on modules. "Basic Algebra is a very interesting and well-written book, and is indeed well suited for the approach to algebra the author intends, and, for that matter, usable for commonplace approaches as well." -MAA Reviews "This is a basic textbook in abstract algebra suitable for a first-year graduate algebra course or perhaps a rather challenging upper level undergraduate course. ... The appendices contain some basic set theory. ... There are many interesting problems, in which a considerable amount of additional material is developed. In particular the theme of coding theory is pursued through many of the chapter exercises and topics such as MacWilliams identity and BCH codes are broached." (Ronald Solomon, Mathematical Reviews, Issue 2007 e) "This book is written as a textbook for students who are learning algebra for the first time. Much of the material corresponds to the contents of a usual algebra course. ... numerous further topics are included which go beyond the level of an undergraduate or even first year graduate student." (H. Mitsch, Monatshefte fur Mathematik, Vol. 156 (4), April, 2009)