The first part explores Galois theory, focusing on related concepts from field theory. The second part discusses the solution of equations by radicals, returning to the general theory of groups for relevant facts, examining equations solvable by radicals and their construction, and concludes with the unsolvability by radicals of the general equation of degree n > 5. 1962 edition.
Author(s): M. M. Postnikov
Series: Dover Books on Mathematics
Edition: Dover Ed
Publisher: Dover Publications
Year: 2004
Language: English
Pages: 119
Contents......Page 3
Foreword......Page 5
Preface......Page 6
I. The Elements of Galois Theory......Page 9
1. Preliminary remarks......Page 11
2. Some important types of extensions......Page 12
3. The minimal polynomial. The structure of simple algebraic extensions......Page 14
4. The algebraic nature of finite extensions......Page 16
5. The structure of composite algebraic extensions......Page 17
6. Composite finite extensions......Page 19
7. The theorem that a composite algebraic extension is simple......Page 22
9. The composition of fields......Page 24
1. The definition of a group......Page 26
2. Subgroups, normal divisors and factor groups......Page 28
3. Homomorphic mappings......Page 31
1. Normal extensions......Page 35
2. Automorphisms of fields. The Galois group......Page 38
3. The order of the Galois group......Page 41
4. The Galois correspondence......Page 45
5. A theorem about conjugate elements......Page 48
6. The Galois group of a normal subfield......Page 49
7. The Galois group of the composition of two fields......Page 51
II. The Solution of Equations by Radicals......Page 53
1. A generalization of the homomorphism theorem......Page 55
2. Normal series......Page 56
3. Cyclic groups......Page 59
4. Solvable and Abelian groups......Page 62
1. Simple radical extensions......Page 68
2. Cyclic extensions......Page 70
3. Radical extensions......Page 75
4. Normal fields with solvable Galois group......Page 78
5. Equations solvable by radicals......Page 81
1. The Galois group of an equation as a group of permutations......Page 83
2. The factorization of permutations into the product of cycles......Page 85
3. Even permutations. The alternating group......Page 89
4. The structure of the alternating and symmetric groups......Page 91
5. An example of an equation with Galois group the symmetric group......Page 95
6. A discussion of the results obtained......Page 99
1. The field of formal power series......Page 102
2. The field of fractional power series......Page 107
3. The Galois group of the general equation of degree n......Page 111
4. The solution of equations of low degree......Page 115
