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Galois Theory
Steven H. Weintraub
Verlag Springer-Verlag, 2009
ISBN 9780387875750 , 220 Seiten
2. Auflage
Format PDF, OL
Kopierschutz Wasserzeichen
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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
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