Abstract Algebra: A COMPREHENSIVE TREATISE - From First Principles to the Research Fronti

Abstract algebra is the mathematics of structure: the language in which symmetry becomes a group, arithmetic a ring, linearity a module, and solvability a question about fields. This comprehensive treatise carries the reader along one continuous road from sets, logic, and the integers to category theory, homological algebra, representation theory, and the threshold of current research. Definitions are earned through the problems they solve, and proofs unfold step by step. Quotients, universal properties, structure theorems, and the celebrated failure of unique factorisation in Z[sqrt(-5)] recur across the book, allowing apparently separate provinces of algebra to reveal their common architecture. Across forty-eight chapters, the volume develops group theory, rings and ideals, modules over principal ideal domains, fields and Galois theory, commutative and noncommutative algebra, characters and representations, Ext, Tor, cohomology, Lie algebras, and Hopf algebras. Worked examples, exercises, computational guidance in GAP and SageMath, selected solutions, a notation index, glossary, and detailed word index support both systematic study and later reference. Written for readers who prefer an argument to a slogan, the book is suited to advanced undergraduates, graduate students, independent learners, and working scientists who require not only the result, but the structure that makes th