Book:Harvey Cohn/Advanced Number Theory

Harvey Cohn: Advanced Number Theory

Published $\text {1962}$, Dover Publications, Inc.

ISBN 0-486-64023-X

Subject Matter

• Number Theory

Contents

Preface

Note: The sections marked with $*$ or $**$ might be omitted in class use if there is a lack of time. (Here the $**$ sections are considered more truly optional.)

Introductory Survey

Diophantine Equations

Motivating Problem in Quadratic Forms

Use of Algebraic Numbers

Primes in Arithmetic Progression

PART 1. BACKGROUND MATERIAL

1. Review of Elementary Number Theory and Group Theory

Number Theoretical Concepts

1. Congruence

2. Unique factorizations

3. The Chinese remainder theorem

4. Structure of reduced residue classes

5. Residue classes for prime powers

Group Theoretic Concepts

6. Abelian groups and subgroups

7. Decomposition into cyclic groups

Quadratic Congruences

8. Quadratic residues

9. Jacobi symbol

$*$2. Characters

1. Definitions

2. Total number of characters

3. Residue classes

4. Resolution modulus

5. Quadratic residue characters

6. Kronecker's symbol and Hasse's congruence

7. Dirichlet's lemma on real characters

3. Some Algebraic Concepts

1. Representation by quadratic forms

2. Use of surds

3. Modules

4. Quadratic integers

5. Hilbert's example

6. Fields

7. Basis of quadratic integers

8. Integral domain

9. Basis of $\sigma_n$

10. Fields of arbitrary degree

4. Basis Theorems

1. Introduction of $n$ dimensions

2. Dirichlet's boxing-in principle

3. Lattices

4. Graphic representation

5. Theorem on existence of basis

6. Other interpretations of the basis construction

7. Lattices of rational integers, canonical basis

8. Sublattices and index concept

9. Applications to modules of quadratic integers

10. Discriminant of a quadratic field

11. Fields of higher degree

$**$5. Further Applications of Basis Theorems

Structure of Finite Abelian Groups

1. Lattice of group relations

2. Need for diagonal basis

3. Elementary divisor theory

4. Basis theorem for abelian groups

5. Simplification of result

Geometric Remarks on Quadratic Forms

6. Successive minima

7. Binary forms

8. Korkine and Zolatareff's example

PART 2. IDEAL THEORY IN QUADRATIC FIELDS

6. Unique Factorization and Units

1. The "missing" factors

2. Indecomposable integers, units, and primes

3. Existence of units in a quadratic field

4. Fundamental units

5. Construction of a fundamental unit

6. Failure of unique factorization into indecomposable integers

7. Euclidean algorithm

8. Occurrence of the Euclidean algorithm

9. Pell's equation

10. Fields of higher degree

7. Unique Factorisation into Ideals

1. Set theoretical notation

2. Definition of ideals

3. Principal ideals

4. Sum of ideals, basis

5. Rules for transforming the ideal

6. Product of ideals, the critical theorem, cancellation

7. "To contain is to divide"

8. Unique factorization

9. Sum and product of factored ideals

10. Two element basis, prime

11. The critical theorem and Hurwitz's 1emma

8. Norms and Ideal Classes

1. Multiplicative property of norms

2. Class structure

3. Minkowski's

4. Norm estimate

9. Class Structure in Quadratic Fields

1. The residue character theorem

2. Primary numbers

3. Determination of principal ideals with given norms

4. Determination of equivalence classes

5. Some imaginary fields

6. Class number unity

7. Units and class calculation of real quadratic fields

8. The famous polynomials $x^2 + x + q$

PART 3. APPLICATIONS OF IDEAL THEORY

$*$10. Class Number Formulas and Primes in Arithmetic Progression

1. Introduction of analysis into number theory

2. Lattice points in

3. Ideal density in complex fields

4. Ideal density in real fields

5. Infinite series, the zeta-function

6. Euler factorization

7. The zeta-function and $L$-series for a field

8. Connection with ideal classes

9. Some simple class numbers

10. Dirichlet $L$-series and primes in arithmetic progression

11. Behavior of the $L$-series, conclusion of proof

12. Weber's theorem on primes in ideal classes

11. Quadratic Reciprocity

1. Rational use of class numbers

2. Results on units

3. Results on class structure

4. Quadratic reciprocity preliminaries

5. The main theorem

6. Kronecker's symbol reappraised

12. Quadratic Forms and Ideals

1. The problem of distinguishing between conjugates

2. The ordered bases of an ideal

3. Strictly equivalent ideals

4. Equivalence classes of quadratic forms

5. The correspondence procedure

6. The correspondence theorem

7. Complete set of classes of quadratic forms

8. Some typical representation problems

$**$13. Compositions, Orders, and Genera

1. Composition of forms

2. Orders, ideals, and forms

3. Genus theory of forms

4. Hilbert's description of genera

$*$CONCLUDING SURVEY

Cyclotomic Fields and Gaussian Sums

Class Fields

Global and Local Viewpoints

Bibliography and Comments

Some Classics Prior to 1900

Some Recent Books (After 1900)

Special References by Chapter

Appendix Tables

I. Minimum Prime Divisors of Numbers Not Divisible by 2, 3 or 5 from 1 to 18,000

II. Power Residues for Primes Less than 100

III. Class Structures of Quadratic Fields of $\sqrt m$ for $m$ Less than 100

Index