First Lectures in Algebra
Why do Normal Subgroups and Ideals Matter?
Series: Springer Asia Pacific Mathematics Series;
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Product details:
- Publisher Springer Nature Singapore
- Date of Publication 14 April 2026
- ISBN 9789819554560
- Binding Hardback
- No. of pages237 pages
- Size 235x155 mm
- Language English
- Illustrations X, 237 p. 1 illus. 700
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Long description:
"
This book is designed as an undergraduate textbook for students in science and engineering, rather than for mathematics majors, yet it maintains full mathematical rigor. It covers groups, rings, modules over rings, finite fields, polynomial rings over finite fields, and error-correcting codes. Even in mathematics departments, undergraduates often wonder why concepts like normal subgroups and ideals matter, and standard textbooks may not provide satisfying answers. This book addresses such questions with both intuition and precision. For example: (1) A normal subgroup is the kernel of a group homomorphism and gives rise to a factor group; a non-normal subgroup does neither. (2) An ideal is a special additive subgroup that serves as the kernel of a ring homomorphism and yields a factor ring; a non-ideal additive subgroup does not. The reader will appreciate the elegant parallelism between these ideas. Key features include:
- A prerequisite chapter that subtly introduces module theory through an elementary presentation of the Euclidean algorithm, accessible even to high school students.
- Recurring use of orbits and clusters, with intuitive illustrations, to clarify the operational meaning of normal subgroups and ideals via homomorphisms.
- Emphasis on proof design patterns, inspired by fields like architecture and software engineering.
- Extensive use of diagrams to support conceptual understanding. Readers are encouraged to draw, compute, design reasoning flows, and then write proofs.
- Complete answers to quizzes and exercises are provided, allowing readers to check their understanding after thoughtful attempts.
Table of Contents:
Chapter 1 Preliminaries.- Chapter 2 Symmetries of Patterns.- Chapter 3 Groups and Orbits.- Chapter 4 Homomorphisms of Groups.- Chapter 5 Symmetric Groups.- Chapter 6 Isomorphism Theorems for Groups.- Chapter 7 Products of Groups.- Chapter 8 Rings and Fields.- Chapter 9 Modules over Rings.- Chapter 10 Ideals.- Chapter 11 Finite Fields and Polynomial Rings.- Chapter 12 Codes and Finite Fields.
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