# Book:Helmut Hasse/Number Theory/Third Edition

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## Helmut Hasse: *Number Theory (3rd Edition)*

Published $\text {1969}$, **Springer**

- ISBN 3-540-42749-X (translated by Horst-Günter Zimmer)

### Subject Matter

### 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