Book:Harvey Cohn/Advanced Number Theory

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