Published: 2024-10-02
Total Pages: 167
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This book is crafted to address the diverse and intricate topics of Abstract Algebra, tailored specifically for students preparing for the CSIR NET(JRF) Mathematical Sciences examination. Each chapter methodically explores essential algebraic structures and theories crucial for a deep understanding of modern algebra. Chapter 1 delves into Group Theory, beginning with foundational concepts of groups and subgroups, and advancing to normal subgroups, quotient groups, and homomorphisms. Key topics include Cayley’s Theorem, class equations, and Sylow theorems, providing a comprehensive overview of group structure and classification. Chapter 2 shifts focus to Ring Theory, exploring rings, subrings, and ideals. It discusses prime and maximal ideals, quotient rings, and various types of domains such as Unique Factorization Domains (UFD), Principal Ideal Domains (PID), and Euclidean Domains, highlighting their properties and interrelationships. Chapter 3 addresses Polynomial Rings, emphasizing their structure and irreducibility criteria. This chapter provides tools for understanding polynomial behavior and factorization. Chapter 4 introduces Field Theory, covering fields, their extensions, and the structures of finite fields. This foundational knowledge sets the stage for advanced topics. Chapter 5 presents Galois Theory, exploring field extensions, automorphisms, and the solvability of equations by radicals. The chapter connects these concepts to broader applications and theoretical implications. This book aims to provide a clear, structured approach to these topics, equipping students with the theoretical insights and problem-solving skills needed for success in the CSIR NET(JRF) examination.