Abstract Algebra: Book Series P-10/24 (Mastering Mathematical Sciences : Complete 24-Book Series for CSIR NET, JRF, GATE, SET & Ph.D. Entrance Exams)

By Dr Jitendra Singh

This book is part of the P-Series , a comprehensive collection of subject-focused volumes meticulously crafted for aspirants of the CSIR-NET (JRF) Mathematical Sciences examination. The P-Series is designed to provide in-depth theoretical insights combined with rigorous practice, helping students master core topics one chapter at a time. This volume focuses on Abstract Algebra , a foundational area in pure mathematics with wide-reaching applications across the mathematical sciences. Each chapter is accompanied by over 100 multiple-choice questions (MCQs) that range in difficulty--from basic conceptual checks to exam-level challenges--mirroring the pattern and expectations of national-level competitive exams. The book is structured as follows: Chapter 1: Group Theory Begins with the definition and properties of groups and subgroups, followed by normal subgroups, quotient groups, and homomorphisms. Key results such as Cayley's Theorem , the Class Equation , and the Sylow Theorems provide a comprehensive understanding of group structure and classification. Chapter 2: Ring Theory Introduces rings, subrings, and ideals, with detailed discussions on prime and maximal ideals . The chapter explores quotient rings and critical domains such as Euclidean Domains (EDs) , Principal Ideal Domains (PIDs) , and Unique Factorization Domains (UFDs) , emphasizing their algebraic properties and relationships. Chapter 3: Polynomial Rings Covers the structure of polynomial rings, with a focus on operations, irreducibility criteria, and factorization techniques--essential tools for understanding algebraic extensions and field construction. Chapter 4: Field Theory Explores the structure and classification of fields and field extensions, including finite fields , algebraic and transcendental extensions . This foundational knowledge paves the way for advanced topics in algebra and number theory. Chapter 5: Galois Theory Investigates the connection between field extensions and group theory through the powerful lens of Galois Theory . Topics include field automorphisms, the Fundamental Theorem of Galois Theory , and the solvability of equations by radicals , alongside applications and structural insights. As a part of the P-Series , this book maintains the series' commitment to academic rigor, clarity of exposition, and exam-oriented preparation. Whether used for conceptual grounding, revision, or practice, this volume aims to empower students with the depth and confidence necessary to succeed in one of India's most competitive mathematical examinations. Stay tuned for other volumes in the P-Series covering Linear Algebra, Real Analysis, Complex Analysis, Topology, and more.

• Format: Hardcover
• Language: English
• ISBN: B0F5NY7MSX
• ISBN13: 9798280570344
• Release Date: April 2025
• Publisher: Independently published
• Length: 275 Pages
• Dimensions: 9.0" x 0.8" x 6.0"