Foundations of Galois Theory (Dover Books on Mathematics)

CONTENTS......Page 4
FOREWORD......Page 6
PREFACE......Page 7
I. THE ELEMENTS OF GALOIS THEORY......Page 10
1. Preliminary remarks......Page 12
2. Some important types of extensions......Page 13
3. The minimal polynomial. The structure of simple algebraicextensions.......Page 15
4. The algebraic nature of finite extensions......Page 17
5. The structure of composite algebraic extensions......Page 18
6. Composite finite extensions......Page 20
7. The theorem that a composite algebraic extension is simple......Page 23
9. The composition of fields......Page 25
1. The definition of a group......Page 27
2. Subgroups, normal divisors and factor groups......Page 29
3. Homomorphic mappings......Page 32
1. Normal extensions......Page 36
2. Automorphisms of fields. The Galois group......Page 39
3. The order of the Galois group......Page 42
4. The Galols correspondence......Page 46
5. A theorem about conjugate elements......Page 49
6. The Galois group of a normal subfield......Page 50
7. The Galois group of the composition of two fields......Page 52
II. THE SOLUTION OF EQUATIONS BYRADICALS......Page 54
1. A generalization of the homomorphism theorem......Page 56
2. Normal series......Page 57
3. Cyclic groups......Page 60
4. Solvable and Abelian groups......Page 63
1. Simple radical extensions......Page 69
2. Cyclic extensions......Page 71
3. Radical extensions......Page 76
4. Normal fields with solvable Galois group......Page 79
5. Equations solvable by radicals......Page 82
1. The Galois group of an equation as a group of permutations......Page 84
2. The factorization of permutations into the product of cycles......Page 86
3. Even permutations. The alternating group......Page 90
4. The structure of the alternating and symmetric groups......Page 92
5. An example of an equation with Galois group the symmetric group......Page 97
6. A discussion of the results obtained......Page 100
1. The field of formal power series......Page 103
2. The field of fractional power series......Page 108
3. The Galois group of the general equation of degree n......Page 112
4. The solution of equations of low degree......Page 116