Rings and Factorization

This textbook is an introduction to the concept of factorization and its application to problems in algebra and number theory.

David Sharpe (Author)

9780521337182, Cambridge University Press

Paperback, published 28 August 1987

124 pages
21.6 x 14 x 0.7 cm, 0.17 kg

This textbook is an introduction to the concept of factorization and its application to problems in algebra and number theory. With the minimum of prerequisites, the reader is introduced to the notion of rings, fields, prime elements and unique factorization. The author shows how concepts can be applied to a variety of examples such as factorizing polynomials, finding determinants of matrices and Fermat's 'two-squares theorem'. Based on an undergraduate course given at the University of Sheffield, Dr Sharpe has included numerous examples which demonstrate how frequently these ideas are useful in concrete, rather than abstract, settings. The book also contains many exercises of varying degrees of difficulty together with hints and solutions. Second and third year undergraduates will find this a readable and enjoyable account of a subject lying at the heart of much of mathematics.

Preface
Introduction
Part I: Rings
1. Introduction
2. Binary operations
3. Definition of a ring
4. Homomorphisms, subrings and isomorphisms
5. Integral domains
6. Fields
7. Residue-class rings
Part II: Factorization
8. Introduction
9. Unique factorization domains
10. Euclidean domains
11. Greatest common divisors
12. Prime elements
13. Euclidean domains are UFDs
14. The two-square theorem
15. Factorization of polynomials
16. An application of UFD's to determinantal identities
Postscript
Partial solutions to exercises
References
Index.

Subject Areas: Algebra [PBF]