这本《数论》由德国Helmut Hasse所著，内容是：The second edition is out of print already six years after its appearance, whereas the first edition was out of print only thirteen years after its appearance. This fact shows that my book meets a still increasing interest from many sides. Since the second edition had been subject to an extensive rewriting and supplementation, it seemed to me that for the third edition I could content myself with making a few corrections and supplementary remarks as well as removing some misprints.

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