《THE THEORY OF ALGEBRAIC NUMBERS》

CHAPTER Ⅰ．Divisibility1

1．The uniqueness of factorization1

2．A general problem5

3．The Gaussian integers7

CHAPTER Ⅱ．The Gaussian Primes12

1．Rational and Gaussian primes12

2．Congruences12

3．Determination of the Gaussian primes16

4．Fermat＇s theorem for Gaussian primes19

CHAPTER Ⅲ．Polynomials over a field22

1．Divisibility properties of polynomials22

2．The Eisenstein irreducibility criterion26

3．Symmetric polynomials31

CHAPTER Ⅳ．Algebraic Number Fields35

1．Numbers algebraic over a field35

2．Extensions of a field37

3．Algebraic and transcendental numbers42

CHAPTER Ⅴ．Bases47

1．Bases and finite extensions47

2．Properties of finite extensions50

3．Conjugates and discriminants52

4．The cyclotomic field55

CHAPTER Ⅵ．Algebraic Integers and Integral Bases58

1．Algebraic integers58

2．The integers in a quadratic field61

3．Integral bases63

4．Examples of integral bases66

CHAPTER Ⅶ．Arithmetic in Algebraic Number Fields71

1．Units and primes71

2．Units in a quadratic field73

3．The uniqueness of factorization76

4．Ideals in an algebraic number field78

CHAPTER Ⅷ．The Fundamental Theorem of Ideal Theory82

1．Basic properties of ideals82

2．The classical proof of the unique factorization theorem86

3．The modern proof92

CHAPTER Ⅸ．Consequences of the Fundamental Theorem96

1．The highest common factor of two ideals96

2．Unique factorization of integers98

3．The problem of ramification101

4．Congruences and norms103

5．Further properties of norms107

CHAPTER Ⅹ．Class-Numbers and Fermat＇s Problem111

1．Class numbers111

2．The Fermat conjecture115

CHAPTER ⅩⅠ．Minkowski＇s Lemma and the Theory of Units125

1．The Minkowski lemma125

2．Applications131

3．The Dirichlet-Minkowski theorem on units132

4．The existence of r independent units134

5．The second part of the proof137

6．The proof completed140

References142

Index143