Galois Theory (eBook)

Steven H. Weintraub (Autor)

eBook Download: PDF

2008 | 2. Auflage
220 Seiten
Springer New York (Verlag)
978-0-387-87575-0 (ISBN)

Infos

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).

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This is a textbook on Galois theory. Galois theory has a well-deserved re- tation as one of the most beautiful subjects in mathematics. I was seduced by its beauty into writing this book. I hope you will be seduced by its beauty in reading it. This book begins at the beginning. Indeed (and perhaps a little unusually for a mathematics text), it begins with an informal introductory chapter, Ch- ter 1. In this chapter we give a number of examples in Galois theory, even before our terms have been properly de?ned. (Needless to say, even though we proceed informally here, everything we say is absolutely correct.) These examples are sort of an airport beacon, shining a clear light at our destination as we navigate a course through the mathematical skies to get there. Then we start with our proper development of the subject, in Chapter 2. We assume no prior knowledge of ?eld theory on the part of the reader. We develop ?eld theory, with our goal being the Fundamental Theorem of Galois Theory (the FTGT). On the way, we consider extension ?elds, and deal with the notions of normal, separable, and Galois extensions. Then, in the penul- mate section of this chapter, we reach our main goal, the FTGT.

Inhalt

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

Produkt-Details

• Erscheint lt. Verlag: 1.1.2009
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Technik
• ISBN-10: 0-387-87575-1 / 0387875751
• ISBN-13: 978-0-387-87575-0 / 9780387875750