Galois theory is one of the most beautiful branches of mathematics. By synthesising the techniques of group theory and field theory it provides a complete answer to the problem of the solubility of polynomials by radicals: that is, the problem of determining when and how a polynomial equation can be solved by repeatedly extracting roots and using elementary algebraic operations. This textbook, based on lectures given over a period of years at Cambridge, is a detailed and thorough introduction to the subject. The work begins with an elementary discussion of groups, fields and vector spaces, and then leads the reader through such topics as rings, extension fields, ruler-and-compass constructions, to automorphisms and the Galois correspondence. By these means, the problem of the solubility of polynomials by radicals is answered; in particular it is shown that not every quintic equation can be solved by radicals. Throughout, Dr Garling presents the subject not as something closed, but as one with many applications. In the final chapters, he discusses further topics, such as transcendence and the calculation of Galois groups, which indicate that there are many questions still to be answered. The reader is assumed to have no previous knowledge of Galois theory. Some experience of modern algebra is helpful, so that the book is suitable for undergraduates in their second or final years. There are over 200 exercises which provide a stimulating challenge to the reader.
Author(s): D. J. H. Garling
Publisher: Cambridge University Press
Year: 1987
Language: English
Pages: viii+167
PREFACE
Part 1: Algebraic preliminaries
1 Groups, fields and vector spaces
1.1 Groups
1.2 Fields
1.3 Vector spaces
2 The axiom of choice, and Zorn's lemma
2.1 The axiom of choice
2.2 Zorn's lemma
2.3 The existence of a basis
3 Rings
3.1 Rings
3.2 Integral domains
3.3 Ideals
3.4 Irreducibles, primes and unique factorization domains
3.5 Principal ideal domains
3.6 Highest common factors
3.7 Polynomials over unique factorization domains
3.8 The existence of maximal proper ideals
3.9 More about fields
Part 2: The theory of fields, and Galois theory
4 Field extensions
4.1 Introduction
4.2 Field extensions
4.3 Algebraic and transcendental elements
4.4 Algebraic extensions
4.5 Monomorphisms of algebraic extensions
5 Tests for irreducibility
5.1 Introduction
5.2 Eisenstein's criterion
5.3 Other methods for establishing irreducibility
6 Ruler-and-compass constructions
6.1 Constructible points
6.2 The angle pi/3 cannot be trisected
6.3 Concluding remarks
7 Splitting fields
7.1 Splitting fields
7.2 The extension of monomorphisms
7.3 Some examples
8 The algebraic closure of a field
8.1 Introduction
8.2 The existence of an algebraic closure
8.3 The uniqueness of an algebraic closure
8.4 Conclusions
9 Normal extensions
9.1 Basic properties
9.2 Monomorphisms and automorphisms
10 Separability
10.1 Basic ideas
10.2 Monomorphisms and automorphisms
10.3 Galois extensions
10.4 Differentiation
10.5 The Frobenius monomorphism
10.6 Inseparable polynomials
11 Automorphisms and fixed fields
11.1 Fixed fields and Galois groups
11.2 The Galois group of a polynomial
11.3 An example
11.4 The fundamental theorem of Galois theory
11.5 The theorem on natural irrationalities
12 Finite fields
12.1 A description of the finite fields
12.2 An example
12.3 Some abelian group theory
12.4 The multiplicative group of a finite field
12.5 The automorphism group of a finite field
13 The theorem of the primitive element
13.1 A criterion in terms of intermediate fields
13.2 The theorem of the primitive element
13.3 An example
14 Cubics and quartics
14.1 Extension by radicals
14.2 The discriminant
14.3 Cubic polynomials
14.4 Quartic polynomials
15 Roots of unity
15.1 Cyclotomic polynomials
15.2 Irreducibility
15.3 The Galois group of a cyclotomic polynomial
16 Cyclic extensions
16.1 A necessary condition
16.2 Abel's theorem
16.3 A sufficient condition
16.4 Kummer extensions
17 Solution by radicals
17.1 Soluble groups: examples
17.2 Soluble groups: basic theory
17.3 Polynomials with soluble Galois groups
17.4 Polynomials which are solvable by radicals
18 Transcendental elements and algebraic independence
18.1 Transcendental elements and algebraic independence
18.2 Transcendence bases
18.3 Transcendence degree
18.4 The tower law for transcendence degree
18.5 Luroth's theorem
19 Some further topics
19.1 Generic polynomials
19.2 The normal basis theorem
19.3 Constructing regular polygons
20 The calculation of Galois groups
20.1 A procedure for determining the Galois group of a polynomial
20.2 The soluble transitive subroups of Sigma_p
20.3 The Galois group of a quintic
20.4 Concluding remarks
Index
Back Cover
