ProofWiki:Books/C.R.J. Clapham/Introduction to Abstract Algebra

C.R.J. Clapham: Introduction to Abstract Algebra

Published 1969, Routledge & Kegan Paul.

ISBN 0 7100 6626 0

Subject Matter

• Integral Domains
• The Integers
• Ring Theory
• Field Theory
• Vector Spaces
• Field Extensions

Contents

Preface

1. Integral Domains

1. Introduction

2. Operations

3. Definition of an Integral Domain

4. Elementary Properties

5. Further Examples of Integral Domains

6. The Residue Classes

2. Ordered and Well-Ordered Integral Domains

7. Order

8. Well-Order

3. The Integers

9. The Principles of Induction

10. Divisibility

11. Highest Common Factor

12. Primes

13. Unique Factorisation

4. Fields

14. Definition of a Field

15. Examples of Fields

16. Subfields

17. The Characteristic of a Field

5. Rings

18. Definition of a Ring

19. Subrings

20. Cosets

21. Ideals

22. Quotient Rings

23. Maximal Ideals

24. Homomorphisms

6. Polynomials and Euclidean Rings

25. Polynomials

26. Divisibility

27. Euclidean Rings

28. Highest Common Factor

29. Irreducible Elements

30. Unique Factorisation

31. Polynomials with Integer Coefficients

7. Vector Spaces

32. Definition of a Vector Space

33. Definition of a Basis

34. Dimension

35. Coordinates

8. Field Extensions

36. The Degree of a Field Extension

37. Roots of a Polynomial

38. Simple Algebraic Extensions

39. A More Sophisticated Approach

40. Constructions with Ruler and Compasses

Exercises

Answers to the Exercises

Index