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The Theory of Algebraic Number Fields

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David Hubert The Theory of Algebraic Number Fields Translated from the German by lain T. Adamson With an Introduction by Franz Lemmermeyer and Norbert Schappacher Table of Contents Translator's Preface V Hilbert's Preface VII Introduction to the English Edition by Franz Lemmermeyer and Norbert Schappacher XXIII 1. The Report XXIII 2. Later Criticism XXV 3. Kummer's Theory XXVIII 4. A Few Noteworthy Details XXXII Part I. The Theory of General Number Fields 1. Algebraic Numbers and Number Fields 3 §1. Number Fields and Their Conjugates 3 §2. Algebraic Integers 4 §3. Norm, Different and Discriminant of a Number. Basis of a Number Field 5 2. Ideals of Number Fields 9 §4. Multiplication and Divisibility of Ideals. Prime Ideals 9 §5. Unique Factorisation of an Ideal into Prime Ideals 11 §6. Forms of Number Fields and Their Contents 14 3. Congruences with Respect to Ideals 17 §7. The Norm of an Ideal and its Properties 17 §8. Fermat's Theorem in Ideal Theory. The Function ip(a) 20 §9. Primitive Roots for a Prime Ideal 22 4. The Discriminant of a Field and its Divisors 25 §10. Theorem on the Divisors of the Discriminant. Lemma on Integral Functions 25 §11. Factorisation and Discriminant of the Fundamental Equation 28 XIV Table of Contents §12. Elements and Different of a Field. Proof of the Theorem on the Divisors of the Discriminant of a Field 30 §13. Determination of Prime Ideals. Constant Numerical Factors of the Rational Unit Form U 31 5. Extension Fields 33 §14. Relative Norms, Differents and Discriminants 33 §15. Properties of the Relative Different and Discriminant 35 §16. Decomposition of an Element of a Field k in an Extension K. Theorem on the Different of the Extension K 38 6. Units of a Field 41 §17. Existence of Conjugates with Absolute Values Satisfying Certain Inequalities 41 §18. Absolute Value of the Field Discriminant 43 §19. Theorem on the Existence of Units 45 §20. Proof of the Theorem on the Existence of Units 49 §21. Fundamental Sets of Units. Regulator of a Field. Independent Sets of Units 51 7. Ideal Classes of a Field 53 §22. Ideal Classes. Finiteness of the Class Number 53 §23. Applications of the Theorem on the Finiteness of the Class Number 54 §24. The Set of Ideal Classes. Strict Form of the Class Concept ... 56 §25. A Lemma on the Asymptotic Value of the Number of All Principal Ideals Divisible by a Given Ideal 56 §26. Determination of the Class Number by the Residue of the Function ((s) at s = 1 60 §27. Alternative Infinite Expansions of the Function ((s) 62 §28. Composition of Ideal Classes of a Field 62 §29. Characters of Ideal Classes. Generalisation of the Function ((s) 64 8. Reducible Forms of a Field 65 §30. Reducible Forms. Form Classes and Their Composition 65 9. Orders in a Field 67 §31. Orders. Order Ideals and Their Most Important Properties ... 67 §32. Order Determined by an Integer. Theorem on the Different of an Integer of a Field 69 §33. Regulär Order Ideals and Their Divisibility Laws 72 §34. Units of an Order. Order Ideal Classes 73 §35. Lattices and Lattice Classes 74 Table of Contents XV Part II. Galois Number Fields 10. Prime Ideals of a Galois Number Field and its Subfields . 79 §36. Unique Factorisation of the Ideals of a Galois Number Field into Prime Ideals 79 §37. Elements, Different and Discriminant of a Galois Number Field 81 §38. Subfields of a Galois Number Field 81 §39. Decomposition Field and Inertia Field of a Prime Ideal 82 §40. A Theorem on the Decomposition Field 83 §41. The Ramification Field of a Prime Ideal 84 §42. A Theorem on the Inertia Field 85 §43. Theorems on the Ramification Group and Ramification Field . 86 §44. Higher Ramification Groups of a Prime Ideal 86 §45. Summary of the Theorems on the Decomposition of a Rational Prime Number p in a Galois Number Field 87 11. The Differents and Discriminants of a Galois Number Field and its Subfields 89 §46. The Differents of the Inertia Field and the Ramification Field 89 §47. The Divisors of the Discriminant of a Galois Number Field ... 90 12. Connexion Between the Arithmetic and Algebraic Properties of a Galois Number Field 93 §48. Galois, Abelian and Cyclic Extension Fields 93 §49. Algebraic Properties of the Inertia Field and the Ramification Field. Representation of the Numbers of a Galois Number Field by Radicals over the Decomposition Field 94 §50. The Density of Prime Ideals of Degree 1 and the Connexion Between this Density and the Algebraic Properties of a Number Field 94 13. Composition of Number Fields 97 §51. The Galois Number Field Formed by the Composition of a Number Field and its Conjugates 97 §52. Compositum of Two Fields Whose Discriminants Are Relatively Prime 98 14. The Prime Ideals of Degree 1 and the Class Concept 101 §53. Generation of Ideal Classes by Prime Ideals of Degree 1 101 15. Cyclic Extension Fields of Prime Degree 105 §54. Symbolic Powers. Theorem on Numbers with Relative Norm 1 105 XVI Table of Contents §55. Fundamental Sets of Relative Units and Proof of Their Existence 106 §56. Existence of a Unit in K with Relative Norm 1 Which is not the Quotient of Two Relatively Conjugate Units 108 §57. Ambig Ideals and the Relative Different of a Cyclic Extension 109 §58. Fundamental Theorem on Cyclic Extensions with Relative Different 1. Designation of These Fields as Class Fields 111 Part III. Quadratic Number Fields 16. Factorisation of Numbers in Quadratic Fields 115 §59. Basis and Discriminant of a Quadratic Field 115 §60. Prime Ideals of a Quadratic Field 116 §61. The Symbol (-) 118 \wJ §62. Units of a Quadratic Field 119 §63. Composition of the Set of Ideal Classes 119 17. Genera in Quadratic Fields and Their Character Sets 121 / ff fr) \ §64. The Symbol ( -i— 121 v w / §65. The Character Set of an Ideal 125 §66. The Character Set of an Ideal Class and the Concept of Genus 126 §67. The Fundamental Theorem on the Genera of Quadratic Fields 127 §68. A Lemma on Quadratic Fields Whose Discriminants are Divisible by Only One Prime 127 §69. The Quadratic Reciprocity Law. A Lemma on the Symbol (^) 128 \ w / §70. Proof of the Relation Asserted in Theorem 100 Between All the Characters of a Genus 131 18. Existence of Genera in Quadratic Fields 133 §71. Theorem on the Norms of Numbers in a Quadratic Field 133 §72. The Classes of the Principal Genus 135 §73. Ambig Ideals 136 §74. Ambig Ideal Classes 136 §75. Ambig Classes Determined by Ambig Ideals 136 §76. Ambig Ideal Classes Containing no Ambig Ideals 138 §77. The Number of All Ambig Ideal Classes 139 §78. Arithmetic Proof of the Existence of Genera 139 Table of Contents XVII §79. Transcendental Representation of the Class Number and an Application that the Limit of a Certain Infinite Product is Positive 140 §80. Existence of Infinitely Many Rational Prime Numbers Modulo Which Given Numbers Have Prescribed Quadratic Residue Characters 142 §81. Existence of Infinitely Many Prime Ideals with Prescribed Characters in a Quadratic Field 144 §82. Transcendental Proof of the Existence of Genera and the Other Results Obtained in Sections 71 to 77 146 §83. Strict Form of the Equivalence and Class Concepts 146 §84. The Fundamental Theorem for the New Class and Genus Concepts 147 19. Determination of the Number of Ideal Classes of a Quadratic Field 149 §85. The Symbol f — 1 for a Composite Number n 149 §86. Closed Form for the Number of Ideal Classes 150 §87. Dirichlet Biquadratic Number Fields 152 20. Orders and Modules of Quadratic Fields 155 §88. Orders of a Quadratic Field 155 §89. Theorem on the Module Classes of a Quadratic Field. Binary Quadratic Forms 155 §90. Lower and Higher Theories of Quadratic Fields 157 Part IV. Cyclotomic Fields 21. The Roots of Unity with Prime Number Exponent l and the Cyclotomic Field They Generate 161 §91. Degree of the Cyclotomic Field of the l-th Roots of Unity; Factorisation of the Prime Number l 161 §92. Basis and Discriminant of the Cyclotomic Field of the l-th Roots of Unity 162 §93. Factorisation of the Rational Primes Distinct from l in the Cyclotomic Field of the l-th Roots of Unity 163 22. The Roots of Unity for a Composite Exponent m and the Cyclotomic Field They Generate 167 §94. The Cyclotomic Field of the m-th Roots of Unity 167 §95. Degree of the Cyclotomic Field of the lh-th Roots of Unity and the Factorisation of the Prime Number l in This Field ... 168 XVIII Table of Contents §96. Basis and Discriminant of the Cyclotomic Field of the lh-th Roots of Unity 168 §97. The Cyclotomic Field of the m-th Roots of Unity. Degree, Discriminant and Prime Ideals of This Field 169 §98. Units of the Cyclotomic Field fc(e2™/m). Definition of the Cyclotomic Units 171 23. Cyclotomic Fields as Abelian Fields 175 §99. The Group of the Cyclotomic Field of the m-th Roots of Unity 175 §100. The General Notion of Cyclotomic Field. The Fundamental Theorem on Abelian Fields 176 §101. A General Lemma on Cyclic Fields 177 §102. Concerning Certain Prime Divisors of the Discriminant of a Cyclic Field of Degree lh 178 §103. The Cyclic Field of Degree u Whose Discriminant is Divisible Only by u and Cyclic Fields of Degree uh and 2h Including U\ and II\ Respectively as Subfields 181 §104. Proof of the Fundamental Theorem on Abelian Fields 184 24.
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