Introduction to Field Theory
Introduction to Field Theory
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USDField Theory is a fascinating branch of algebra, with many interesting applications, and its central result, the Fundamental Theorem of Galois Theory, is by any standards one of the really important theorems of mathematics. This book brings the reader from the basic definitions to important results and applications, and introduces him to the spirit and some of the techniques of abstract algebra. It is addressed to undergraduates in pure mathematics and presupposes only a little knowledge of elementary group theory. Chapter I develops the elementary properties of rings and fields including the notions of characteristic, prime fields and various types of homomorphisms. In Chapter II extension fields and various ways of classifying them are studies. Chapter III gives an exposition of the Galois Theory, following Artin's approach, and Chapter IV provides a wide variety of applications of the preceding theory. For the second edition Dr Adamson has improved the exposition in places, made corrections and updated the references.
Reviews & endorsements
Review of the hardback: 'This is an attractive book on field theory and Galois theory…it is very clearly written, with many examples, and the exercises are good…an excellent introduction.' American Mathematical Monthly
Product details
September 1982Paperback
9780521286589
192 pages
203 × 127 × 11 mm
0.22kg
Available
Table of Contents
- Preface
- Part I: Elementary Definitions
- 1. Rings and fields
- 2. Elementary properties
- 3. Homomorphisms
- 4. Vector spaces
- 5. Polynomials
- 6. Higher polynomial rings
- rational functions
- Part II: Extensions of fields
- 7. Elementary properties
- 8. Simple extensions
- 9. Algebraic extensions
- 10. Factorisation of polynomials
- 11. Splitting fields
- 12. Algebraically closed fields
- 13. Separable extensions
- Part III: Galois theory
- 14. Automorphisms of fields
- 15. Normal extensions
- 16. The fundamental theorem of Galois Theory
- 17. Norms and traces
- 18. The primitive element theorem
- Lagrange's theorem
- 19. Normal bases
- Part IV: Applications
- 20. Finite fields
- 21. Cyclotomic extensions
- 22. Cyclotomic extensions of the rational number field
- 23. Cyclic extensions
- 24. Wedderburns' theorem
- 25. Ruler-an-compasses constructions
- 26. Solution by radicals
- 27. Generic polynomials.
