Book:Helmut Hasse/Number Theory/Third Edition

Subject Matter
Number Theory

Contents

 * Preface to the English Edition (Saarbrücken, May 1978)


 * Preface to the First Edition (Berlin, February 1949)


 * Preface to the Second Edition (Hamburg, November 1962)


 * Preface to the Third Edition (Honolulu, Hawaii, Spring 1969)

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 $$\mod m$$
 * 1. General Facts Concerning Direct Products and Direct Sums
 * 2. Direct Decomposition of the Residue Class Ring $$\mod m$$ and of the Reduced Residue Class Group $$\mod m$$
 * 3. The Structure of the Additive Group of the Residue Class Ring $$\mod m$$
 * 4. On the Structure of the Residue Class Ring $$\mod p^\mu$$
 * 5. The Structure of the Reduced Residue Class Group $$\mod p^\mu$$
 * 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 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. 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 Abelian Operator Group
 * 3. The Field of $$n$$th Roots of Unity over a $$p$$-adic Number Field
 * 4. The Structure of the One-Unit Group in the Equal-Charaeteristic Case with Finite Residue Class Field
 * 5. The Structure of the One-Unit Group in the $$\mathfrak p$$-adic Case
 * 6. Construction of a system of fundamental One-units in the $$\mathfrak p$$-adic Case
 * 7. The One-Unit Group for Special $$\mathfrak p$$-adic Number Fields
 * 8. Comparison of the Basis Representation of the Multiplicative Group in the $$\mathfrak p$$-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 Complete 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-Archimedean 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. Characterisations 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 Numbers
 * 4. Quotient Representation
 * Function Fields
 * Constant Fields, Constant Extensions


 * Chapter 23. Congruences
 * 1. Ordinary Congruence
 * 2 Multiplicative 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 $$\mathfrak p$$. 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 Qnadratic 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. Minkowski's Lattice-Point Theorem
 * 3. Application to Convex Bodies within the Norm-one Hypersurface
 * 4. Consequences of the Discriminant Estimate
 * Function Fields


 * Index of Names


 * Subject Index