# Book:Harvey Cohn/Advanced Number Theory

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## Harvey Cohn:

## Harvey Cohn: *Advanced Number Theory*

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

- ISBN 0-486-64023-X.

### Subject Matter

### 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

- Number Theoretical Concepts

- Group Theoretic Concepts
- 6. Abelian groups and subgroups
- 7. Decomposition into cyclic groups

- Group Theoretic Concepts

- Quadratic Congruences
- 8. Quadratic residues
- 9. Jacobi symbol

- Quadratic Congruences

**$*$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

- Structure of Finite Abelian Groups

- Geometric Remarks on Quadratic Forms
- 6. Successive minima
- 7. Binary forms
- 8. Korkine and Zolatareff's example

- Geometric Remarks on Quadratic Forms

#### 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**