THE THEORY OF ALGEBRAIC NUMBERSPDF电子书下载

CHAPTER Ⅰ．Divisibility 1

1．The uniqueness of factorization 1

2．A general problem 5

3．The Gaussian integers 7

CHAPTER Ⅱ．The Gaussian Primes 12

1．Rational and Gaussian primes 12

2．Congruences 12

3．Determination of the Gaussian primes 16

4．Fermat＇s theorem for Gaussian primes 19

CHAPTER Ⅲ．Polynomials over a field 22

1．Divisibility properties of polynomials 22

2．The Eisenstein irreducibility criterion 26

3．Symmetric polynomials 31

CHAPTER Ⅳ．Algebraic Number Fields 35

1．Numbers algebraic over a field 35

2．Extensions of a field 37

3．Algebraic and transcendental numbers 42

CHAPTER Ⅴ．Bases 47

1．Bases and finite extensions 47

2．Properties of finite extensions 50

3．Conjugates and discriminants 52

4．The cyclotomic field 55

CHAPTER Ⅵ．Algebraic Integers and Integral Bases 58

1．Algebraic integers 58

2．The integers in a quadratic field 61

3．Integral bases 63

4．Examples of integral bases 66

CHAPTER Ⅶ．Arithmetic in Algebraic Number Fields 71

1．Units and primes 71

2．Units in a quadratic field 73

3．The uniqueness of factorization 76

4．Ideals in an algebraic number field 78

CHAPTER Ⅷ．The Fundamental Theorem of Ideal Theory 82

1．Basic properties of ideals 82

2．The classical proof of the unique factorization theorem 86

3．The modern proof 92

CHAPTER Ⅸ．Consequences of the Fundamental Theorem 96

1．The highest common factor of two ideals 96

2．Unique factorization of integers 98

3．The problem of ramification 101

4．Congruences and norms 103

5．Further properties of norms 107

CHAPTER Ⅹ．Class-Numbers and Fermat＇s Problem 111

1．Class numbers 111

2．The Fermat conjecture 115

CHAPTER ⅩⅠ．Minkowski＇s Lemma and the Theory of Units 125

1．The Minkowski lemma 125

2．Applications 131

3．The Dirichlet-Minkowski theorem on units 132

4．The existence of r independent units 134

5．The second part of the proof 137

6．The proof completed 140

References 142

Index 143