Basic Algebra

Basic Algebra and Advanced Algebra systematically develop concepts and tools in algebra that are vital to every mathematician, whether pure or applied, aspiring or established. Basic Algebra and Advanced Algebra systematically develop concepts and tools in algebra that are vital to every mathematician, whether pure or applied, aspiring or established. Together, the two books give the reader a global view of algebra and its role in mathematics as a whole. Key topics and features of Basic Algebra: *Linear algebra and group theory build on each other continually *Chapters on modern algebra treat groups, rings, fields, modules, and Galois groups, with emphasis on methods of computation throughout *Three prominent themes recur and blend together at times: the analogy between integers and polynomials in one variable over a field, the interplay between linear algebra and group theory, and the relationship between number theory and geometry *Many examples and hundreds of problems are included, along with a separate 90-page section giving hints or complete solutions for most of the problems *The exposition proceeds from the particular to the general, often providing examples well before a theory that incorporates them; includes blocks of problems that introduce additional topics and applications for further study *Applications to science and engineering (e.g., the fast Fourier transform, the theory of error-correcting codes, the use of the Jordan canonical form in solving linear systems of ordinary differential equations, and constructions of interest in mathematical physics) appear in sequences of problems Basic Algebra presents the subject matter in a forward-looking way that takes into account its historical development. It is suitable as a text in a two-semester advanced undergraduate or first-year graduate sequence in algebra, possibly supplemented by some material from Advanced Algebra at the graduate level. It requires of the reader only familiarity with matrix algebra, an understanding of the geometry and reduction of linear equations, and an acquaintance with proofs. 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.