Algebra: Third Edition

Main subject category: • AlgebraChapter Headings: • 1. Sets, Functions, and Integers • 2. Groups • 3. Rings • 4. Universal Constructions • 5. Modules • 6. Vector Spaces • 7. Matrices • 8. Special Fields • 9. Determinants and Tensor Products • 10. Bilinear and Quadratic Forms • 11. Similar Matrices and Finite Abelian Groups • 12. Structure of Groups • 13. Galois Theory • 14. Lattices • 15. Categories and Adjoint Functors • 16. Multilinear Algebra • Appendix: Affine and Projective Spaces • Bibliography • IndexThis book presents modern algebra from first principles and is accessible to undergraduates or graduates. It combines standard materials and necessary algebraic manipulations with general concepts that clarify meaning and importance.This conceptual approach to algebra starts with a description of algebraic structures by means of axioms chosen to suit the examples, for instance, axioms for groups, rings, fields, lattices, and vector spaces. This axiomatic approach ‒ emphasized by Hilbert and developed in Germany by Noether, Artin, Van der Waerden, et al., in the 1920s ‒ was popularized for the graduate level in the 1940s and 1950s to some degree by the authors' publication of A Survey of Modern Algebra. The present book presents the developments from that time to the first printing of this book. This third edition includes corrections made by the authors. This book presents modern algebra from first principles and is accessible to undergraduates or graduates. It combines standard materials and necessary algebraic manipulations with general concepts that clarify meaning and importance. This conceptual approach to algebra starts with a description of algebraic structures by means of axioms chosen to suit the examples, for instance, axioms for groups, rings, fields, lattices, and vector spaces. This axiomatic approach--emphasized by Hilbert and developed in Germany by Noether, Artin, Van der Waerden, et al., in the 1920s--was popularized for the graduate level in the 1940s and 1950s to some degree by the authors' publication of A Survey of Modern Algebra. The present book presents the developments from that time to the first printing of this book. This third edition includes corrections made by the authors. Presents modern algebra from first principles. This title combines standard materials and necessary algebraic manipulations with general concepts that clarify meaning and importance. It presents a conceptual approach to algebra that starts with a description of algebraic structures by means of axioms chosen to suit the examples. This third edition examines the fundamentals of algebra.