The Theory of Algebraic Number Fields

Cover
Title page
Translator's Preface
Hilbert's Preface
Introduction to the English Edition by Franz Lemmermeyer and Norbert Schappacher
1. The Report
2. Later Criticism
3. Kummer's Theory
4. A Few Noteworthy Details
Part I. The Theory of General Number Fields
1. Algebraic Numbers and Number Fields
§1. Number Fields and Their Conjugates
§2. Algebraic Integers
§3. Norm, Different and Discriminant of a Number. Basis of a Number Field
2. Ideals of Number Fields
§4. Multiplication and Divisibility of Ideals. Prime Ideals
§5. Unique Factorisation of an Ideal into Prime Ideals
§6. Forms of Number Fields and Their Contents
3. Congruences with Respect to Ideals
§7. The Norm of an Ideal and its Properties
§8. Fermat's Theorem in Ideal Theory. The Function φ(a)
§9. Primitive Roots for a Prime Ideal
4. The Discriminant of a Field and its Divisors
§10. Theorem on the Divisors of the Discriminant. Lemma on Integral Functions
§11. Factorisation and Discriminant of the Fundamental Equation
§12. Elements and Different of a Field. Proof of the Theorem on the Divisors of the Discriminant of a Field
§13. Determination of Prime Ideals. Constant Numerical Factors of the Rational Unit Form U
5. Extension Fields
§14. Relative Norms, Differents and Discriminants
§15. Properties of the Relative Different and Discriminant
§16. Decomposition of an Element of a Field k in an Extension K. Theorem on the Different of the Extension K
6. Units of a Field
§17. Existence of Conjugates with Absolute Values Satisfying Certain Inequalities
§18. Absolute Value of the Field Discriminant
§19. Theorem on the Existence of Units
§20. Proof of the Theorem on the Existence of Units
§21. Fundamental Sets of Units. Regulator of a Field. Independent Sets of Units
7. Ideal Classes of a Field
§22. Ideal Classes. Finiteness of the Class Number
§23. Applications of the Theorem on the Finiteness of the Class Number
§24. The Set of Ideal Classes. Strict Form of the Class Concept
§25. A Lemma on the Asymptotic Value of the Number of All Principal Ideals Divisible by a Given Ideal
§26. Determination of the Class Number by the Residue of the Function ζ(s) at s=1
§27. Alternative Infinite Expansions of the Function ζ(s)
§28. Composition of Ideal Classes of a Field
§29. Characters of Ideal Classes. Generalisation of the Function ζ(s)
8. Reducible Forms of a Field
§30. Reducible Forms. Form Classes and Their Composition
9. Orders in a Field
§31. Orders. Order Ideals and Their Most Important Properties
§32. Order Determined by an Integer. Theorem on the Different of an Integer of a Field
§33. Regular Order Ideals and Their Divisibility Laws
§34. Units of an Order. Order Ideal Classes
§35. Lattices and Lattice Classes
Part II. Galois Number Fields
10. Prime Ideals of a Galois Number Field and its Subfields
§36. Unique Factorisation of the Ideals of a Galois Number Field into Prime Ideals
§37. Elements, Different and Discriminant of a Galois Number Field
§38. Subfields of a Galois Number Field
§39. Decomposition Field and Inertia Field of a Prime Ideal
§40. A Theorem on the Decomposition Field
§41. The Ramification Field of a Prime Ideal
§42. A Theorem on the Inertia Field
§43. Theorems on the Ramification Group and Ramification Field
§44. Higher Ramification Groups of a Prime Ideal
§45. Summary of the Theorems on the Decomposition of a Rational Prime Number p in a Galois Number Field
11. The Differents and Discriminants of a Galois Number Field and its Subfields
§46. The Differents of the Inertia Field and the Ramification Field
§47. The Divisors of the Discriminant of a Galois Number Field
12. Connexion Between the Arithmetic and Aigebraic Properties of a Galois Number Field
§48. Galois, Abelian and Cyclic Extension Fields
§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
§50. The Density of Prime Ideals of Degree 1 and the Connexion Between this Density and the Algebraic Properties of a Number Field
13. Composition of Number Fields
§51. The Galois Number Field Formed by the Composition of a Number Field and its Conjugates
§52. Compositum of Two Fields Whose Discriminants Are Relatively Prime
14. The Prime Ideals of Degree 1 and the Class Concept
§53. Generation of Ideal Classes by Prime Ideals of Degree 1
15. Cyclic Extension Fields of Prime Degree
§54. Symbolic Powers. Theorem on Numbers with Relative Norm 1
§55. Fundamental Sets of Relative Units and Proof of Their Existence
§56. Existence of a Unit in K with Relative Norm 1 Which is not the Quotient of Two Relatively Conjugate Units
§57. Ambig Ideals and the Relative Different of a Cyclic Extension
§58. Fundamental Theorem on Cyclic Extensions with Relative Different 1. Designation of These Fields as Class Fields
Part III. Quadratic Number Fields
16. Factorisation of Numbers in Quadratic Fields
§59. Basis and Discriminant of a Quadratic Field
§60. Prime Ideals of a Quadratic Field
§61. The Symbol (a/w)
§62. Units of a Quadratic Field
§63. Composition of the Set of Ideal Classes
17. Genera in Quadratic Fields and Their Character Sets
§64. The Symbol (n,m/w)
§65. The Character Set of an Ideal
§66. The Character Set of an Ideal Class and the Concept of Genus
§67. The Fundamental Theorem on the Genera of Quadratic Fields
§68. A Lemma on Quadratic Fields Whose Discriminants are Divisible by Only One Prime
§69. The Quadratic Reciprocity Law. A Lemma on the Symbol (n,m/w)
§70. Proof of the Relation Asserted in Theorem 100 Between All the Characters of a Genus
18. Existence of Genera in Quadratic Fields
§71. Theorem on the Norms of Numbers in a Quadratic Field
§72. The Classes of the Principal Genus
§73. Ambig Ideals
§74. Ambig Ideal Classes
§75. Ambig Classes Determined by Ambig Ideals
§16. Ambig Ideal Classes Containing no Ambig Ideals
§17. The Number of All Ambig Ideal Classes
§78. Arithmetic Proof of the Existence of Genera
§79. Transcendental Representation of the Class Number and an Application that the Limit of a Certain Infinite Product is Positive
§80. Existence of Infinitely Many Rational Prime Numbers Modulo Which Given Numbers Have Prescribed Quadratic Residue Characters
§81. Existence of Infinitely Many Prime Ideals with Prescribed Characters in a Quadratic Field
§82. Transcendental Proof of the Existence of Genera and the Other Results Obtained in Sections 71 to 77
§83. Strict Form of the Equivalence and Class Concepts
§84. The Fundamental Theorem for the New Class and Genus Concepts
19. Determination of the Number of Ideal Classes of a Quadratic Field
§85. The Symbol (a/n) for a Composite Number n
§86. Closed Form for the Number of Ideal Classes
§87. Dirichlet Biquadratic Number Fields
20. Orders and Modules of Quadratic Fields
§88. Orders of a Quadratic Field
§89. Theorem on the Module Classes of a Quadratic Field. Binary Quadratic Forms
§90. Lower and Higher Theories of Quadratic Fields
Part IV. Cyclotomic Fields
21. The Roots of Unity with Prime Number Exponent l and the Cyclotomic Field They Generate
§91. Degree of the Cyclotomic Field of the l-th Roots of Unit y; Factorisation of the Prime Number l
§92. Basis and Discriminant of the Cyclotomic Field of the l-th Roots of Unity
§93. Factorisation of the Rational Primes Distinct from l in the Cyclotomic Field of the l-th Roots of Unity
22. The Roots of Unity for a Composite Exponent m and the Cyclotomic Field They Generate
§94. The Cyclotomic Field of the m-th Roots of Unity
§95. Degree of the Cyclotomic Field of the l^h-th Roots of Unity and the Factorisation of the Prime Number l in This Field
396. Basis and Discriminant of the Cyclotomic Field of the l^h-th Roots of Unity
§97. The Cyclotomic Field of the rn-th Roots of Unity. Degree, Discriminant and Prime Ideals of This Field
§98. Units of the Cyclotomic Field k(e^{2iπ/m}). Definition of the Cyclotomic Units
23. Cyclotomic Fields as Abelian Fields
§99. The Group of the Cyclotomic Field of the m-th Roots of Unity
§100. The General Notion of Cyclotomic Field. The Fundamental Theorem on Abelian Fields
§101. A General Lemma on Cyclic Fields
§102. Concerning Certain Prime Divisors of the Discriminant of a Cyclic Field of Degree l^h
§103. The Cyclic Field of Degree u Whose Discriminant is Divisible Only by u and Cyclic Fields of Degree u^h and 2^h Including U₁ and II₁ Respectively as Subfields
§104. Proof of the Fundamental Theorem on Abelian Fields
24. The Root Numbers of the Cyclotomic Field of the l-th Roots of Unity
§105. Definition and Existence of Normal Bases
§106. Abelian Fields of Prime Degree l and Discriminant p^{l-1}. Root Numbers of This Field
§107. Characteristic Properties of Root Numbers
§108. Factorisation of the l-th Power of a Root Number in the Field of the l-th Roots of Unity
§109. An Equivalence for the Prime Ideals of Degree 1 in the Field of the l-th Roots of Unity
§110. Construction of An Normal Bases and Root Numbers
§111. The Lagrange Normal Basis and the Lagrange Root Number
§112. The Characteristic Properties of the Lagrange Root Number
25. The Reciprocity Law for l-th Power Residues Between a Rational Number and a Number in the Field of l-th Roots of Unity
§113. The Power Character of a Number and the Symbol (a/p)
§114. A Lemma on the Power Character of the l-th Power of the Lagrange Root Number
§115. Proof of the Reciprocity Law in the Field k(ζ) Between a Rational Number and an Arbitrary Number
26. Determination of the Number of Ideal Classes in the Cyclotomic Field of the rn-th Roots of Unity
§116. The Symbol (a/L)
§117. The Expression for the Class Number of the Cyclotomic Field of the m-th Roots of Unity
§118. Derivation of the Expressions for the Class Number of the Cyclotomic Field k(e^{2Iπ/m})
§119. The Existence of Infinitely Many Rational Primes with a Prescribed Residue Modulo a Given Number
§120. Representation of All the Units of the Cyclotomic Field by Cyclotomic Units
27. Applications of the Theory of Cyclotomic Fields to Quadratic Fields
§121. Generation of the Units of Real Quadratic Fields by Cyclotomic Units
§122. The Quadratic Reciprocity Law
§123. Imaginary Quadratic Fields with Prime Discriminant
§124. Determination of the Sign of the Gauss Sum
Part V. Kummer Number Fields
28. Factorisation of the Numbers of the Cyclotomic Field in a Kummer Field
§125. Definition of Kummer Fields
§126. The Relative Discriminant of a Kummer Field
§127. The Symbol {μ/m}
§128. The Prime Ideals of a Kummer Field
29. Norm Residues and Non-residues of a Kummer Field
§129. Definition of Norm Residues and Non-residues
§130. Theorem on the Number of Norm Residues. Ramification Ideals
§131. The Symbol {ν,μ/m}
§132. Some Lemmas on the Symbol {ν,μ/ι} and Norm Residues Modulo the Prime Ideal ι
§133. Use of the Symbol {ν,μ/m} to Distinguish Norm Residues and Non-residues
30. Existence of Infinitely Many Prime Ideals with Prescribed Power Characters in a Kummer Field
§134. The Limit of a Certain Infinite Product
§135. Prime Ideals of the Cyclotomic Field k(ζ) with Prescribed Power Characters
31. Regular Cyclotomic Fields
§136. Definition of Regular Cyclotomic Fields, Regular Prime Numbers and Regular Kummer Fields
§137. A Lemma on the Divisibility by l of the First Factor of the Class Number of k(e^{2iπ/l})
§138. A Lemma on the Units of the Cyclotomic Field k(e^{2iπ/l}) When l Does Not Divide the Numerators of the First (l-3)/2 Bernoulli Numbers
§139. A Criterion for Regular Prime Numbers
§140. A Special Independent Set of Units in a Regular Cyclotomic Field
§141. A Characteristic Property of the Units of a Regular Cyclotomic Field
§142. Primary Numbers in Regular Cyclotomic Fields
32. Ambig Ideal Classes and Genera in Regular Kummer Fields
§143. Unit Bundles in Regular Cyclotomic Fields
§144. Ambig Ideals and Ambig Ideal Classes of a Regular Kummer Field
§145. Class Bundles in Regular Kummer Fields
§146. Two General Lemmas on Fundamental Sets of Relative Units of a Cyclic Extension of Odd Prime Number Degree
§147. Ideal Classes Determined by Ambig Ideals
§148. The Set of All Ambig Ideal Classes
§149. Character Sets of Numbers and Ideals in Regular Kummer Fields
§150. The Character Set of an Ideal Class and the Notion of Genus
§151. Upper Bound for the Degree of the Class Bundle of All Ambig Classes
§152. Complexes in a Regular Kummer Field
§153. An Upper Bound for the Number of Genera in a Regular Kummer Field
33. The l-th Power Reciprocity Law in Regular Cyclotomic Fields
§154. The l-th Power Reciprocity Law and the Supplementary Laws
§155. Prime Ideals of First and Second Kind in a Regular Cyclotomic Field
§156. Lemmas on Prime Ideals of the First Kind in Regular Cyclotomic Fields
§157. A Particular Case of the Reciprocity Law for Two Ideals
§158. The Existence of Certain Auxiliary Prime Ideals for Which the Reciprocity Law Holds
§159. Proof ofthe First Supplementary Lawofthe Reciprocity Law
§160. Proof of the Reciprocity Law for Any Two Prime Ideals
§161. Proof of the Second Supplementary Law for the Reciprocity Law
34. The Number of Genera in a Regular Kummer Field
§162. A Theorem on the Symbol {ν,μ/m}
§163. The Fundamental Theorem on the Genera of a Regular Kummer Field
§164. The Classes of the Principal Genus in a Regular Kummer Field
§165. Theorem on the Relative Norms of Numbers in a Regular Kummer Field
35. New Foundation of the Theory of Regular Kummer Fields
§166. Essential Properties of the Units of a Regular Cyclotomic Field
§167. Proof of a Property of Primary Numbers for Prime Ideals of the Second Kind
§168. Proof of the Reciprocity ... Where One of the Two Prime Ideals is of the Second Kind
§169. A Lemma About the Product etc Where m Runs Over All Prime Ideals Distinct from l
§170. The Symbol {ν,μ} and the Reciprocity Law Between Any Two Prime Ideals
§171. Coincidence of the Symbols {ν,μ} and {ν,μ/ι}
36. The Diophantine Equation α^m + β^m + γ^m = 0
§172. The Impossibility of the Diophantine Equation α^l + β^l + γ^l = 0 for a Regular Prime Number Exponent l
§173. Further Investigations on the Impossibility of the Diophantine Equation α^l + β^l + γ^l = 0
References
List of Theorems and Lemmas
Index