Book:Helmut Hasse/Number Theory/Third Edition

Helmut Hasse: Number Theory (3rd Edition)

Published $\text {1969}$, Springer

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

Number Theory
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

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

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