Introduction to Field Theory

This book brings the reader from the basic definitions to important results and applications, and introduces some of the techniques of abstract algebra.

Iain T. Adamson (Author)

9780521286589, Cambridge University Press

Paperback, published 2 September 1982

192 pages
20.3 x 12.7 x 1.1 cm, 0.22 kg

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

Field 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.

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.

Subject Areas: Algebra [PBF]