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. The Root Numbers of the Cyclotomic Field of the Z-th Roots of Unity 187 §105. Definition and Existence of Normal Bases 187 §106. Abelian Fields of Prime Degree l and Discriminant pl~l. Root Numbers of This Field 188 §107. Characteristic Properties of Root Numbers 188 §108. Factorisation of the Z-th Power of a Root Number in the Field of the Z-th Roots of Unity 192 §109. An Equivalence for the Prime Ideals of Degree 1 in the Field of the l-th. Roots of Unity 193 §110. Construction of All Normal Bases and Root Numbers 194 §111. The Lagrange Normal Basis and the Lagrange Root Number 195 §112. The Characteristic Properties of the Lagrange Root Number 195

25. The Reciprocity Law for l-th. Power Residues Between a Rational Number and a Number in the Field of Z-th Roots of Unity 199 §113. The Power Character of a Number and the Symbol l — > ... 199 §114. A Lemma on the Power Character of the Z-th Power of the Lagrange Root Number 200 §115. Proof of the Reciprocity Law in the Field k(() Between a Rational Number and an Arbitrary Number 202 Table of Contents XIX

26. Determination of the Number of Ideal Classes in the Cyclotomic Field of the m-th Roots of Unity 207 §116. The Symbol j] 207 §117. The Expression for the Class Number of the Cyclotomic Field of the m-th Roots of Unity 208 §118. Derivation of the Expressions for the Class Number of the Cyclotomic Field fc(e27ri/m) 211 §119. The Existence of Infinitely Many Rational Primes with a Prescribed Residue Modulo a Given Number 213 §120. Representation of All the Units of the Cyclotomic Field by Cyclotomic Units 215

27. Applications of the Theory of Cyclotomic Fields to Quadratic Fields 217 §121. Generation of the Units of Real Quadratic Fields by Cyclotomic Units 217 §122. The Quadratic Reciprocity Law 217 §123. Imaginary Quadratic Fields with Prime Discriminant 219 §124. Determination of the Sign of the Gauss Sum 220

Part V. Kummer Number Fields

28. Factorisation of the Numbers of the Cyclotomic Field in a Kummer Field 225 §125. Definition of Kummer Fields 225 §126. The Relative Discriminant of a Kummer Field 226 §127. The Symbol {-} 229 §128. The Prime Ideals of a Kummer Field 230

29. Norm Residues and Non-residues of a Kummer Field 233 §129. Definition of Norm Residues and Non-residues 233 §130. Theorem on the Number of Norm Residues. Ramification Ideals 233 §131. The Symbol ( —) 240 L ro J §132. Some Lemmas on the Symbol < -y- > and Norm Residues Modulo the Prime Ideal [ 243 §133. Use of the Symbol < —— > to Distinguish Norm Residues and Non-residues 248 XX Table of Contents

30. Existence of Infinitely Many Prime Ideals with Prescribed Power Characters in a Kummer Field .... 253 §134. The Limit of a Certain Infinite Product 253 §135. Prime Ideals of the Cyclotomic Field k(() with Prescribed Power Characters 254

31. Regulär Cyclotomic Fields 257 §136. Definition of Regulär Cyclotomic Fields, Regulär Prime Numbers and Regulär Kummer Fields 257 §137. A Lemma on the Divisibility by l of the First Factor of the Class Number of fc(e27"/') 257 §138. A Lemma on the Units of the Cyclotomic Field k(e2lTl/l) When / Does Not Divide the Numerators of the First \{l — 3) Bernoulli Numbers 259 §139. A Criterion for Regulär Prime Numbers 262 §140. A Special Independent Set of Units in a Regulär Cyclotomic Field 264 §141. A Characteristic Property of the Units of a Regulär Cyclotomic Field 265 §142. Primary Numbers in Regulär Cyclotomic Fields 266

32. Ambig Ideal Classes and Genera in Regulär Kummer Fields 269 §143. Unit Bundles in Regulär Cyclotomic Fields 269 §144. Ambig Ideals and Ambig Ideal Classes of a Regulär Kummer Field 270 §145. Class Bundles in Regulär Kummer Fields 270 §146. Two General Lemmas on Fundamental Sets of Relative Units of a Cyclic Extension of Odd Prime Number Degree 271 §147. Ideal Classes Determined by Ambig Ideals 273 §148. The Set of All Ambig Ideal Classes 280 §149. Character Sets of Numbers and Ideals in Regulär Kummer Fields 282 §150. The Character Set of an Ideal Class and the Notion of Genus 284 §151. Upper Bound for the Degree of the Class Bündle of All Ambig Classes 285 §152. Complexes in a Regulär Kummer Field 286 §153. An Upper Bound for the Number of Genera in a Regulär Kummer Field 287

33. The Z-th Power Reciprocity Law in Regulär Cyclotomic Fields 289 §154. The ^-th Power Reciprocity Law and the Supplementary Laws . . 289 §155. Prime Ideals of First and Second Kind in a Regulär Cyclotomic Field 290 Table of Contents XXI

§156. Lemmas on Prime Ideals of the First Kind in Regulär Cyclotomic Fields 293 §157. A Particular Case of the Reciprocity Law for Two Ideals .... 296 §158. The Existence of Certain Auxiliary Prime Ideals for Which the Reciprocity Law Holds 298 §159. Proofofthe First Supplementary Law oftheReciprocity Law .. 300 §160. Proof of the Reciprocity Law for Any Two Prime Ideals .... 301 §161. Proof of the Second Supplementary Law for the Reciprocity Law 303

34. The Number of Genera in a Regulär Kummer Field 305 §162. A Theorem on the Symbol j —} 305 §163. The Fundamental Theorem on the Genera of a Regulär Kummer Field 306 §164. The Classes of the Principal Genus in a Regulär Kummer Field 308 §165. Theorem on the Relative Norms of Numbers in a Regulär Kummer Field 309

35. New Foundation of the Theory of Regulär Kummer Fields 313 §166. Essential Properties of the Units of a Regulär Cyclotomic Field 313 §167. Proof of a Property of Primary Numbers for Prime Ideals of the Second Kind 315 §168. Proof of the Reciprocity Law Where One of the Two Prime Ideals is of the Second Kind 318 §169. A Lemma About the Product JJ [ — j Where tu Runs Over All Prime Ideals Distinct from [ 321 §170. The Symbol {v, fx) and the Reciprocity Law Between Any Two Prime Ideals 324 §171. Coincidence of the Symbols {v, ß} and < —!— > 325

36. The Diophantine Equation am + ßm + -f1 = 0 327 §172. The Impossibility of the Diophantine Equation al + ßl + 7' = 0 for a Regulär Prime Number Exponent l . . . 327 §173. Further Investigations on the Impossibility of the Diophantine Equation al + ßl + jl = 0 332

References 335

List of Theorems and Lemmas 345

Index 347