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.