First lectures in algebra why do normal subgroups and ideals matter? Shuichi Yukita

Format:

Book

Author/Creator:

Yukita, Shuichi, author.

Series:

Springer Asia Pacific mathematics series ; 3091-2563 v. 10

Springer Asia Pacific mathematics series 3091-2563 volume 10

Language:

English

Subjects (All):

Algebra.

algebra.

Physical Description:

1 online resource

Place of Publication:

Singapore Springer [2026]

Summary:

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

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

Notes:

Includes bibliographical references and index

Online resource; title from PDF title page (SpringerLink, viewed April 27, 2026)

ISBN:

9789819554577

9819554578

OCLC:

1587160734

Access Restriction:

Restricted for use by site license