A Concrete Approach to Abstract Algebra: From the Integers to the Insolvability of the Quintic, Second Edition provides a primer and reference on abstract algebra for readers whose interests lie in mathematics and information and physical sciences. Adopting the unique 'rings first' approach, the work provides a gentle transition into abstract structures that will make abstract algebra more natural to interested readers.
In addition to introducing the major concepts of modern algebra, the book covers numerous applications which are intended to illustrate the concepts and convince the reader of the utility and relevance of algebra today. This Second Edition features 40% new or revised content, including complete and self-contained proofs of the fundamental theorems of algebra and the Insolvability of the Quintic, and new coverage of commutative rings and linear transformations.
- Offers an extraordinarily diverse reference of the algebraic field providing foundational progression through algebraic concepts suitable for newcomers and experts alike
- Demonstrates in simple language-using multiple examples and exact proofs-how most concepts within abstract algebra are actually tools used to solve difficult, but well-known problems
- Employs a gradual approach to build on relatively familiar material (integers, polynomials)
- Explores more abstract topics while providing the classical approach of introducing groups first as automorphisms
- Supports both prospective graduate students as well as prospective teachers
1. What this book is about and who this book is for 2. Proof and Intuition 3. The Integers 4. The Rational Numbers and the Real Numbers 5. The Complex Numbers 6. The Integers 7. Group Theory: A First Look 8. Roots and Factorization of Polynomials over the integrals 9. Polynomials over arbitrary fields 10. Difference Functions and Partial Fractions 11. An Introduction to Linear Algebra and Vector Spaces 12. Degrees and Galois Groups of Field Extensions 13. Geometric Constructions 14. Group Theory: Solvable and Symmetric Groups 15. Insolvability of the Quintic 16. Values of Trigonometric Functions 17. Additional Topics
Jeffrey Bergen (DePaul, Chicago), received his B.S. in Mathematics from Brooklyn College in 1976. He received his M.S. in 1977 and Ph.D. in 1981 from the University of Chicago. His DePaul career began in 1981, where he continues to do research in the branch of abstract algebra known as noncommutative ring theory. His research has received external support from the English Speaking Union, the National Science Foundation, and the National Security Agency. He has given lectures in 7 countries and co-authored papers with 16 mathematicians around the world. In 2001, he received the Excellence in Teaching Award from the College of Liberal Arts and Sciences and, in 2007, received their Cortelyou-Lowery Award for Excellence.