A Book of Abstract Algebra, 2nd edition – Charles C. Pinter

Diving into abstract algebra for the first time can be intimidating; it certainly was for me. However, it wasn’t until after I left school that I discovered this book. This book is a delightful foundation in abstract algebra for a reader who has never studied algebra before. Pinter breaks very intangible topics into short, bite-sized sections on topics that have very clear, almost visual examples. He keeps the number of theorems in each section short, and uses the exercises to help the reader build an intuition and motivation for more abstract topics. I think the best feature of this book is the quality of carefully chosen exercises that show just how much abstract algebra shows up everywhere, even outside of mathematics. Many algebra books don’t take the time to really draw out the Cayley Diagrams to play with, or give lots of exploratory exercises to understand symmetric groups, or really make sure the reader can a way to “touch” these topics before diving into concepts like isomorphism. The book would be fantastic to teach from, and is suitable for an undergraduate, or perhaps even very advanced high school text.  

Review

Prerequisites

basic high school algebra, some logic background


Topics Covered

  • Operations
  • Groups/Subgroups
    • generators and defining relations
  • Permutations
  • Isomorphism
  • Cyclic Groups
  • Partitions and Equivalence Relations
  • Cosets
  • Homomorphisms
  • Quotient Groups
  • Sylow Subgroups/ Sylow’s Theorem
  • Rings
  • Ideals
  • Quotient Rings
  • Integral Domains
  • Polynomials – Rings, Factoring, and Substitutions
  • Extensions of Fields
  • Vector Spaces
  • Intro to Galois Theory

Attributes

Difficulty 2
Proof Quality 4
Readability 5
Self-Study 5
Good for teaching? 5
Quality of Exercises 5