Front Cover	1
	Foundations of Galois Theory	4
	Copyright Page	5
	Table of Contents	6
	Foreword	8
	Preface	9
	PART I: THE ELEMENTS OF GALOIS THEORY	12
	CHAPTER 1. THE ELEMENTS OF FIELD THEORY	14
	1. Preliminary remarks	14
	2. Some important types of extensions	15
	3. The minimal polynomial. The structure of simple algebraic extensions	17
	4. The algebraic nature of finite extensions	19
	5. The structure of composite algebraic extensions	20
	6. Composite finite extensions	22
	7. The theorem that a composite algebraic extension is simple	25
	8. The field of algebraic numbers	27
	9. The composition of fields	27
	CHAPTER 2. NECESSARY FACTS FROM THE THEORY OF GROUPS	29
	1. The definition of a group	29
	2. Subgroups, normal divisors and factor groups	31
	3. Homomorphic mappings	34
	CHAPTER 3. GALOIS THEORY	38
	1. Normal extensions	38
	2. Automorphisms of fields. The Galois group	41
	3. The order of the Galois group	44
	4. The Galois correspondence	48
	5. A theorem about conjugate elements	51
	6. The Galois group of a normal subfield	52
	7. The Galois group of the composition of two fields	54
	PART II: THE SOLUTION OF EQUATIONS BY RADICALS	56
	CHAPTER 1. ADDITIONAL FACTS FROM THE GENERAL THEORY OF GROUPS	58
	1. A generalization of the homomorphism theorem	58
	2. Normal series	59
	3. Cyclic groups	62
	4. Solvable and Abelian groups	65
	CHAPTER 2. EQUATIONS SOLVABLE BY RADICALS	71
	1. Simple radical extensions	71
	2. Cyclic extensions	73
	3. Radical extensions	78
	4. Normal fields with solvable Galois group	81
	5. Equations solvable by radicals	84
	CHAPTER 3. THE CONSTRUCTION OF EQUATIONS SOLVABLE BY RADICALS	86
	1. The Galois group of an equation as a group of permutations	86
	2. The factorization of permutations into the product of cycles	88
	3. Even permutations. The alternating group	92
	4. The structure of the alternating and symmetric groups	94
	5. An example of an equation with Galois group the symmetric group	98
	6. A discussion of the results obtained	102
	CHAPTER 4. THE UNSOLVABILITY BY RADICALS OF THE GENERAL EQUATION OF DEGREE n ? 5	105
	1. The field of formal power series	105
	2. The field of fractional power series	110
	3. The Galois group of the general equation of degree n	114
	4. The solution of equations of low degree	118