Abstract Algebra: A Gentle Introduction advantages a trend in mathematics textbook publishing towards smaller, less expensive and brief introductions to primary courses. The authors move away from the ‘everything for everyone’ approach so common in textbooks. Instead, they provide the reader with coverage of numerous algebraic topics to cover the most important areas of abstract algebra.
Through a careful selection of topics, supported by interesting applications, the authors Intend the book to be used for a one-semester course in abstract algebra. It is suitable for an introductory course in for mathematics majors. The text is also very suitable for education majors
who need to have an introduction to the topic.
As textbooks go through various editions and authors employ the suggestions of numerous well-intentioned reviewers, these book become larger and larger and subsequently more expensive. This book is meant to counter that process. Here students are given a "gentle introduction," meant to provide enough for a course, yet also enough to encourage them toward future study of the topic.
As the subtitle implies, those seeking a standard undergraduate text in abstract algebra should look elsewhere. The authors provide readers with a very brief introduction to some of the central structures of algebra: groups, rings, fields, and vector spaces. As an example of the textâ€™s brevity, its treatment of groups consists of definitions, examples, and a discussion of subgroups and cosets that culminates in LaGrangeâ€™s theorem. There is no mention of group homomorphisms, normal subgroups, or quotient groups. Nonetheless, various applications of the subject not often addressed in traditional texts are treated within this work. It appears that the intent is to provide enough content for readers to comprehend these applications. Just enough elementary number theory is presented to allow a discussion of the RSA cryptosystem. Sufficient material on finite fields is given for a discussion of Latin squares and the Diffie-Hellman public key exchange. Adequate linear algebra topics foster a discussion of Hamming codes. This text will be suitable for an algebra-based course introducing students to abstract mathematical thought or an algebra course with an emphasis on applications.
--D. S. Larson, Gonzaga University, Choice magazine 2016
Elementary Number Theory
Primes and factorization
Theorems of Fermat and Euler
Definition of a group
Examples of groups
Cosets and Lagrange's Theorem
Definition of a ring
Subrings and ideals
Definition and basic properties of a field
Number of elements in a finite field
How to construct finite fields
Properties of finite fields
Polynomials over finite fields
Orthogonal latin squares
Di□e/Hellman key exchange
De nition and examples
Basic properties of vector spaces
Polynomials over the real and complex numbers
Hints and Partial Solutions to Selected Exercises