Weintraub Galois Theory

2. Auflage 2009
ISBN: 978-0-387-87575-0
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 220 Seiten

Reihe: Universitext

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. 

Key topics and features of this second edition: 
- Approaches Galois theory from the linear algebra point of view, following Artin, 
- Presents a number of applications of Galois theory, including symmetric functions, finite fields, cyclotomic fields, algebraic number fields, solvability of equations by radicals, and the impossibility of solution of the three geometric problems of Greek antiquity. 

Review from the first edition: The text offers the standard material of classical field theory and Galois theory, though in a remarkably original, unconventional and comprehensive manner … . the book under review must be seen as a highly welcome and valuable complement to existing textbook literature … . It comes with its own features and advantages … it surely is a perfect introduction to this evergreen subject. The numerous explaining remarks, hints, examples and applications are particularly commendable … just as the outstanding clarity and fullness of the text. (Zentralblatt MATH, Vol. 1089 (15), 2006) Steven H. Weintraub is a Professor of Mathematics at Lehigh University and the author of seven books. This book grew out of a graduate course he taught at Lehigh. He is also the author of Algebra: An Approach via Module Theory (with W. A. Adkins).

Autoren/Hrsg.

Weintraub, Steven H.

Inhaltsverzeichnis

1;Contents;6
2;Preface to the First Edition;8
3;Preface to the Second Edition;11
4;1 Introduction to Galois Theory;12
4.1;1.1 Some Introductory Examples;12
5;2 Field Theory and Galois Theory;18
5.1;2.1 Generalities on Fields;18
5.2;2.2 Polynomials;22
5.3;2.3 Extension Fields;26
5.4;2.4 Algebraic Elements and Algebraic Extensions;29
5.5;2.5 Splitting Fields;33
5.6;2.6 Extending Isomorphisms;35
5.7;2.7 Normal, Separable, and Galois Extensions;36
5.8;2.8 The Fundamental Theorem of Galois Theory;40
5.9;2.9 Examples;48
5.10;2.10 Exercises;51
6;3 Development and Applications of Galois Theory;56
6.1;3.1 Symmetric Functions and the Symmetric Group;56
6.2;3.2 Separable Extensions;65
6.3;3.3 Finite Fields;67
6.4;3.4 Disjoint Extensions;71
6.5;3.5 Simple Extensions;77
6.6;3.6 The Normal Basis Theorem;80
6.7;3.7 Abelian Extensions and Kummer Fields;84
6.8;3.8 The Norm and Trace;90
6.9;3.9 Exercises;93
7;4 Extensions of the Field of Rational Numbers;99
7.1;4.1 Polynomials in Q[X];99
7.2;4.2 Cyclotomic Fields;103
7.3;4.3 Solvable Extensions and Solvable Groups;107
7.4;4.4 Geometric Constructions;111
7.5;4.5 Quadratic Extensions of Q;117
7.6;4.6 Radical Polynomials and Related Topics;122
7.7;4.7 Galois Groups of Extensions of Q;132
7.8;4.8 The Discriminant;138
7.9;4.9 Practical Computation of Galois Groups;141
7.10;4.10 Exercises;147
8;5 Further Topics in Field Theory;153
8.1;5.1 Separable and Inseparable Extensions;153
8.2;5.2 Normal Extensions;161
8.3;5.3 The Algebraic Closure;165
8.4;5.4 Infinite Galois Extensions;170
8.5;5.5 Exercises;181
9;6 Transcendental Extensions;183
9.1;6.1 General Results;183
9.2;6.2 Simple Transcendental Extensions;191
9.3;6.3 Plane Curves;195
9.4;6.4 Exercises;201
10;A Some Results from Group Theory;204
10.1;A.1 Solvable Groups;204
10.2;A.2 p-Groups;208
10.3;A.3 Symmetric and Alternating Groups;209
11;B A Lemma on Constructing Fields;214
12;C A Lemma from Elementary Number Theory;216
13;Index;218