數論（簡體書）

《數論》內容簡介：In spite of the fact that nowadays there are quite a few books on algebraic numbertheory available to the mathematical community, there seems to be still a strongneed for a fundamental work like Hasses ,Zahlentheorie. This impression iscorroborated by the great number of inquiries the editor received about the dateof appearance of the English translation of Hasses book. One main reason for theunbroken interest in this book lies probably in its vivid presentation of the divisor-theoretic approach to algebraic number theory, an approach which was developedby Hasses former teacher Hensel and further expanded by Hasse himself. Hassedoes not content himself with a mere presentation of the number-theoretic mate-rial, but he motivates the basic ideas and questions, comments on them in detail,and points out their connections with neighboring branches of mathematics. In preparing the Eng本書是Springer經典數學系列之一，是一部深層次講述數論的書籍，從賦值理論的角度全面闡釋數論知識。自從1949年該理論面世以來，許多人在這方面做出了很大貢獻。作者這本書自有特別之處，提出了局部-整體理論（亦Hasse原理），並且給出了完美的證明，作為數論方面的學者，本書既是一本教科書，也是一本不可或缺的案頭參考書。目次：有理數域中的算術基礎：素數分解；可除性；一致性；模m的留數類環結構；二次留數；值域論：基本概念；離散值域中的算術；值域的完備化；離散值域的完備化；具有完全留數類域的完全離散值域同構類型；純超越擴張的離散賦值延伸；有限代數擴張的完全域賦值延伸；完全阿基米德值域的同構類；完全離散值域的有限代數擴展的結構；具有素特徵完全留數類域的完全離散值域乘法群結構；指數函數；有限代數擴張的不完全域賦值延伸；代數數域中的算術基礎：賦值完全體系和有理數域算術之間的聯繫；有限代數擴張的賦值完全體系延伸；代數數域的素點及其完備化；分解為素因子，整數和可分性；一致性；因子的多重性；差積和判別式；二次數域；割圓域；類數；逼近定理和判別式估計。
讀者對象：數學專業的研究生，科研人員，數論研究專家。

作者：（德國）哈塞（Helmut Hasse）

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 mod m and of the Reduced Residue Class Group rood ra
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 Ringrood m
4. On the Structure of the Residue Class Ring rood pu
5. The Structure of the Reduced Residue Class Group rood pu
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 Jaeobi Symbol
6. The Quadratic Reciprocity Law as Product Formula for the
Hilbert Symbol
7. Special Cases of Dirichlcts Theorem on Prime Numbers in
Reduced Residue Classes
Function Field

Part Ⅱ. The Theory of Valued Flel&
Chapter 6. The Fundamental Concepts Regarding Valuations
1. The Definition of a Valuation; Equivalent Valuations
2. Approximation Independence and Multiplicative 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. Witts Vector Calculus
5. Construction of the General p-adie Field
6. The Unequal-Characteristic Case
7. Isomorphic Residue Systems in the Case of Characteristic
8. The Isomorphic Residue Systems for a Rational Function Field
9. The Equal-Characteristic Case with Characteristic
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 Uuramified 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 Multiplieative Group of a Complete Discrete
Valued Field with Perfect Residue Class Field of Prime Charac teristic
1. Reduction to the One-Unit Group and its Fundamental Chain of Subgroups
2. The One-Unit Group as an Abelian Operator Group
3. The Field of nth Roots of Unity over a p-adie Number Field
4. The Structure of the One-Unit Group in the Equal-Charac teristic Case with Finite Residue Class Field
5. The Structure of the One-Unit Group in the p-adie Case
6. Construction of a System of Fundamental One-Units in the p-adic Case
7. The One-Unit Group for Special p-adic Number Fields
8. Comparison of the Basis Representation of the Multiplicative Group in the Cadic 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
Chapter 17. The Exponential Function, the Logarithm, and Powers in a Complete Non-Archimedean Valued Field of Characteristic
1. Integral Power Series in One Indeterminate over an Arbitrary Field
2. Integral Power Series in One Variable in a Complete Non-Archimedean Valued Field
3. Convergence
4. Functional Equations and Mutual Relations
5. The Discrete Case
6. The Equal-Characteristic Case with Characteristic
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 Part
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 Com pletions
Function Fields
Chapter 22. Decomposition into Prime Divisors, Integrality, and Divisibility
1. The Canonical Homomorphism of the Multiplicative Groupinto the Divisor Group
2. Embedding of Divisibility Theory under a Finite-Algebraic Extension
3. Algebraic Characterization of Integral Algebraic Numbers .
4. Quotient Representation
Function Fields
Constant Fields, Constant Extensions
Chapter 23. Congruences
1. Ordinary Congruence
2. Multiplicaive 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 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
I. Generation in the Large and in the Small
2. The Decomposition Law
3. Disoriminants, 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. Cyclotomic 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. Minkowskis Lattice.Point Theorem
3. Application to Convex Bodice within the Norm-one Hyper-surface
4. Consequences of the Diacriminant Estimate
Function Fields
Index o！ Names
Subject Index