Part Ⅰ. The Foundations of Arithmetic In the Rational Number Field
Chapter 1. Prime Decomposition
Function Fields
Chapter 2. Divisibility
Function Fields
Chapter 3. Congruences
Function Fields
The Theory of Finite Fields
Chapter 4. The Structure of the Residue Class Ring rood m and of the Reduced Residue Class Group rood m
1. General Facts Concerning Direct Products and Direct Sums
2. Direct Decomposition of the Residue Class Ring rood m and of the Reduced Residue Class Group rood m
3. The Structure of the Additive Group of the Residue Class Ring rood m
4. On the Structure of the Residue Class Ring rood pμ
5. The Structure of the Reduced Residue Class Group rood pμ
Function Fields
Chapter 5. Quadratic Residues
1. Theory of the Characters of a Finite Abelian Group
2. Residue Class Characters and Numerical Characters mod m
3. The Basic Facts Concerning Quadratic Residues
4. The Quadratic Reciprocity Law for the Legendre Symbol
5. The Quadratic Reciprocity Law for the Jacobi Symbol
6. The Quadratic Reciprocity Law as Product Formula for the Hilbert Symbol
7. Special Cases of Dirichlet's Theorem on Prime Numbers in Reduced Residue Classes
Function Field
Part Ⅱ. The Theory of Valued Fields
Chapter 6. The Fundamental Concepts Regarding Valuations
1. The Definition of a Valuation; Equivalent Valuations
2. Approximation Independence and Muitiplicative Independence of Valuations
3. Valuations of the Prime Field
4. Value Groups and Residue Class Fields
Function Fields
Chapter 7. Arithmetic in a Discrete Valued Field
Divisors from an Ideal-Theoretic Standpoint
Chapter 8. The Completion of a Valued Field
Chapter 9. The Completion of a Discrete Valued Field. The p-adic Number Fields
Function Fields
Chapter 10. The Isomorphism Types of Complete Discrete Valued Fields with Perfect Residue Class Field
1. The Multiplicative Residue System in the Case of Prime Characteristic
2. The Equal-Characteristic Case with Prime Characteristic
3. The Multiplicative Residue System in the p-adic Number Field
4. Witt's Vector Calculus
5. Construction of the General p-adic Field
6. The Unequal-Characteristic Case
7. Isomorphic Residue Systems in the Case of Characteristic 0 .
8. The Isomorphic Residue Systems for a Rational Function Field
9. The Equal-Characteristic Case with Characteristic 0
Chapter 11. Prolongation of a Discrete Valuation to a Purely Transcendental Extension
Chapter 12. Prolongation of the Valuation of a Complete Field to a Finite- Algebraic Extension
1. The Proof of Existence
2. The Proof of Completeness
3. The Proof of Uniqueness
Chapter 13. The Isomorphism Types of Complete Archimedean Valued Fields
Chapter 14. The Structure of a Finite-Algebraic Extension of a Complete Discrete Valued Field
1. Embedding of the Arithmetic
2. The Totally Ramified Case
3. The Unramified Case with Perfect Residue Class Field
4. The General Case with Perfect Residue Class Field
5. The General Case with Finite Residue Class Field
Chapter 15. The Structure of the Multiplicative Group of a Complete Discrete Valued Field with Perfect Residue Class Field of Prime Characteristic
1. Reduction to the One-Unit Group and its Fundamental Chain of Subgroups
2. The One-Unit Group as an Ahelian Operator Group
3. The Field of nth Roots of Unity over a 10-adie Number Field
4. The Structure of the One-Unit Group in the Equal-Characteristic Case with Finite Residue Class Field
5. The Structure of the One-Unit Group in the p-adic Case
6. Construction of a System of Fundamental One-Units in the p-adie Case
7. The One-Unit Group for Special p-adic Number Fields . . .
8. Comparison of the Basis Representation of the Multiplicative Group in the p-adic Case and the Archimedean Case
Chapter 16. The Tamely Ramified Extension Types of a Complete Discrete Valued Field with Finite Residue Class Field of Characteristic p
Chapter 17. The Exponential Function, the Logarithm, and Powers in a Com- plete Non-Archimedean Valued Field of Characteristic 0
1. Integral Power Series in One Indeterminate over an Arbitrary Field
2. Integral Power Series in One Variable in a Complete Non-Archi- medean Valued Field
3. Convergence
4. Functional Equations and Mutual Relations
5. The Discrete Case
6. The Equal-Characteristic Case with Characteristic 0
Chapter 18. Prolongation of the Valuation of a Non-Complete Field to a Finite-Algebraic Extension
1. Representations of a Separable Finite-Algebraic Extension over an Arbitrary Extension of the Ground Field
2. The Ring Extension of a Separable Finite-Algebraic Extension by an Arbitrary Ground Field Extension, or the Tensor Product of the Two Field Extensions
3. The Characteristic Polynomial
4. Supplements for Inseparable Extensions
5. Prolongation of a Valuation
6. The Discrete Case
7. The Archimedean Case
Part Ⅲ. The Foundations of Arithmetic in Algebraic Number Fields
Chapter 19. Relations Between the Complete System of Valuations and the Arithmetic of the Rational Number Field
1. Finiteness Properties
2. Characterizations in Divisibility Theory
3. The Product Formula for Valuations
4. The Sum Formula for the Principal Parts
Function Fields
The Automorphisms of a Rational Function Field
Chapter 20. Prolongation of the Complete System of Valuations to a Finite-Algebraic Extension
Function Fields
Concluding Remarks
Chapter 21. The Prime Spots of an Algebraic Number Field and their Completions
Function Fields
Chapter 22. Decomposition into Prime Divisors, Integrality, and Divisibility
1. The Canonical Homomorphism of the Multiplicative Group into the Divisor Group
2. Embedding of Divisibility Theory under a Finite-Algebraic Extension
3. Algebraic Characterization of Integral Algebraic Numbcrs
4. Quotient Representation
Function Fields
Constant Fields, Constant Extensions
Chapter 23. Congruences
1. Ordinary Congruence
2. Multip]icative Congruence
Function Fields
Chapter 24. The Multiples of a Divisor
1. Field Bases
2. The Ideal Property, Ideal Bases
3. Congruences for Integral Elements
4. Divisors from the Ideal-Theoretic Standpoint
5. Further Remarks Concerning Divisors and Ideals
Function Fields
Constant Fields for p. Characterization of Prime Divisors by
Homomorphisms. Decomposition Law under an Algebraic
Constant Extension
The Rank of the Module of Multiples of a Divisor
Chapter 25. Differents and Discriminants
1. Composition Formula for the Trace and Norm. The Divisor Trace
2. Definition of the Different and Discriminant
3. Theorems on Differents and Discriminants in the Small
4. The Relationship Between Differents and Discriminants in the Small and in the Large
5. Theorems on Differents and Discriminants in the Large
6. Common Inessential Discriminant Divisors
7. Examples
Function Fields
The Number of First-Degree Prime Divisors in the Case of a Finite Constant Field Differentials
The Riemann-Roch Theorem and its Consequences
Disclosed Algebraic Function Fields
Chapter 26. Quadratic Number Fields
1. Generation in the Large and in the Small
2. The Decomposition Law
3. Discriminants, Integral Bases
4. Quadratic Residue Characters of the Discriminant of an Arbitrary Algebraic Number Field
5. The Quadratic Number Fields as Class Fields
6. The Hilbert Symbol as Norm Symbol
7. The Norm Theorem
8. A Necessary Condition for Principal Divisors. Genera
Chapter 27. Cyclotemic Fields
1. Generation
2. The Decomposition Law
3. Discriminants, Integral Bases
4. The Quadratic Number Fields as Subfields of Cyclotomic Fields
Chapter 28. Units
1. Preliminaries
2. Proofs
3. Extension
4. Examples and Applications
Chapter 29. The Class Number
1. Finiteness of the Class Number
2. Consequences
3. Examples and Applications Function Fields
Chapter 30. Approximation Theorems and Estimates of the Discriminant
1. The Most General Requirements on Approximating Zero
2. Minkowaki's Lattice-Point Theorem
3. Application to Convex Bodies within the Norm-one Hyper-surface
4. Consequences of the Discriminant Estimate Function Fields
Index of Names
Subject Index
