Preface ........................................................ ix
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
2.9. Examples ............................................... 37
2.10.Exercises .............................................. 39
3. Development and Applications of Galois Theory ............... 45
3.1. Symmetric Functions and the Symmetric Group ............ 45
3.2. Separable Extensions ................................... 51
3.3. Finite Fields .......................................... 54
3.4. Disjoint Extensions .................................... 57
3.5. Simple Extensions ...................................... 63
3.6. The Normal Basis Theorem ............................... 66
3.7. Abelian Extensions and Kummer Fields ................... 70
3.8. The Norm and Trace ..................................... 76
3.9. Exercises .............................................. 79
4. Extensions of the Field of Rational Numbers ................. 85
4.1. Polynomials in Q[X] .................................... 85
4.2. Cyclotomic Fields ...................................... 89
4.3. Solvable Extensions and Solvable Groups ................ 93
4.4. Geometric Constructions ................................ 97
4.5. Quadratic Extensions of Q ............................. 103
4.6. Radical Polynomials and Related Topics ................ 108
4.7. Galois Groups of Extensions of Q ...................... 118
4.8. The Discriminant ...................................... 124
4.9. Practical Computation of Galois Groups ................ 127
4.10.Exercises ............................................. 133
5. Further Topics in Field Theory ............................. 139
5.1. Separable and Inseparable Extensions .................. 139
5.2. Normal Extensions ..................................... 147
5.3. The Algebraic Cslosure ................................ 151
5.4. Infinite Galois Extensions ............................ 156
5.5. Exercises ............................................. 167
A. Some Results from Group Theory ............................. 169
A.l. Solvable Groups ....................................... 169
A.2. p-Groups .............................................. 173
A.3. Symmetric and Alternating Groups ...................... 174
В. A Lemma on Constructing Fields ............................. 179
С. A Lemma from Elementary Number Theory ...................... 181
Index ......................................................... 183
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