This book can be roughly divided into two parts (which may correspond to 2 semesters in a 2-course sequence):
- The first part deals with foundational examples that historically were the inspiration for the development of abstract algebra : complex numbers, modular numbers (with applications to cryptography), and polynomials, symmetries, and permutation groups. Also included are reviews of sets and functions, as well as summation notation. We don't really talk about groups,rings, etc. in general, because our purpose is to give students a concrete examples of the abstract concepts that will come later.
- The second part gets more into abstractions, including elementary properties of groups, rings, and fields (with most of the emphasis on groups). We do not go into much theoretical depth (for example, the Sylow theorems are not mentioned), because our intended audience includes pending secondary teachers as well as scientists/engineers who can benefit from some exposure to the algebraic way of thinking. We also talk about applications such as group actions, cryptography (including elliptic curve cryptography) and error correction coding.
We hesitate to give a recommended syllabus, because the book is designed to be customized to the instructor's individual needs. Chapters may be readily omitted if students are already familiar with the material. Some chapters ("Preliminaries" and "Sigma Notation") are remedial. Other chapters cover topics that are often covered in courses in discrete mathematics, such as sets, functions, and equivalence classes.
An Instructor's Supplement is available to course instructors upon request (see the "Contact us" page). The supplement contains solutions to many (but not all) exercises. There are also test sample questions that can be used for exams.
The LaTeX source code is also available upon request.