Galois theory / Steven H. Weintraub.

Format:

Book

Author/Creator:

Weintraub, Steven H.

Series:

Universitext

Language:

English

Subjects (All):

Galois theory.

Physical Description:

xi, 211 pages : illustrations ; 24 cm.

Edition:

Second edition.

Place of Publication:

New York : Springer, [2009]

Summary:

The book discusses classical Galois theory in considerable generality, treating fields of characteristic zero and of positive characteristic with consideration of both separable and inseparable extensions, but with a particular emphasis on algebraic extensions of the field of rational numbers. While most of the book is concerned with finite extensions, it discusses algebraic closure and infinite Galois extensions, and concludes with a new chapter on transcendental extensions.

Contents:

1 Introduction to Galois Theory 1

1.1 Some Introductory Examples 1

2 Field Theory and Galois Theory 7

2.1 Generalities on Fields 7

2.2 Polynomials 11

2.3 Extension Fields 15

2.4 Algebraic Elements and Algebraic Extensions 18

2.5 Splitting Fields 22

2.6 Extending Isomorphisms 24

2.7 Normal, Separable, and Galois extensions 25

2.8 The Fundamental Theorem of Galois Theory 29

3 Development and Applications of Galois Theory 45

3.1 Symmetric Functions and the Symmetric Group 45

3.2 Separable Extensions 54

3.3 Finite Fields 56

3.4 Disjoint Extensions 60

3.5 Simple Extensions 66

3.6 The Normal Basis Theorem 69

3.7 Abelian Extensions and Kummer Fields 73

3.8 The Norm and Trace 79

4 Extensions of the Field of Rational Numbers 89

4.1 Polynomials in Q[X] 89

4.2 Cyclotomic Fields 93

4.3 Solvable Extensions and Solvable Groups 97

4.4 Geometric Constructions 101

4.5 Quadratic Extensions of Q 107

4.6 Radical Polynomials and Related Topics 112

4.7 Galois Groups of Extensions of Q 122

4.8 The Discriminant 128

4.9 Practical Computation of Galois Groups 131

5 Further Topics in Field Theory 143

5.1 Separable and Inseparable Extensions 143

5.2 Normal Extensions 151

5.3 The Algebraic Closure 155

5.4 Infinite Galois Extensions 160

6 Transcendental Extensions 173

6.1 General Results 173

6.2 Simple Transcendental Extensions 181

6.3 Plane Curves 185

A Some Results from Group Theory 195

A.1 Solvable Groups 195

A.2 p-Groups 199

A.3 Symmetric and Alternating Groups 200

B A Lemma on Constructing Fields 205

C A Lemma from Elementary Number Theory 207.

Notes:

Previous ed.: 2005.

Includes index.

ISBN:

9780387875743

0387875743

0387875751

9780387875750

OCLC:

298595680