Book:H.E. Rose/A Course in Number Theory/Second Edition

Subject Matter

 * Number Theory

Contents

 * Preface to Second Edition (Bristol, January $1994$)


 * Preface to First Edition (Bristol, February $1987$)


 * 1 Divisibility
 * $1$ The Euclidean algorithm and unique factorization
 * $2$ Prime numbers
 * $3$ Problems $1$


 * 2 Multiplicative Functions
 * $1$ The Möbius and Euler functions
 * $2$ Average order
 * $3$ Problems $2$


 * 3 Congruence Theory
 * $1$ Definitions and linear congruences
 * $2$ Nonlinear congruences and the theorems of Euler, Lagrange, and Chevalley
 * $3$ Local versus global considerations
 * $4$ Computation modulo $n$
 * $5$ Problems $3$


 * 4 Quadratic Residues
 * $1$ The Legendre symbol
 * $2$ Quadratic reciprocity
 * $3$ Some further topics
 * $4$ Problems $4$


 * 5 Algebraic Topics
 * $1$ Algebraic numbers and integers
 * $2$ Primitive roots
 * $3$ Characters
 * $4$ Problems $5$


 * 6 Sums of Squares and Gauss Sums
 * $1$ Sums of squares
 * $2$ Gauss and Jacobi sums
 * $3$ The sign of the quadratic Gauss sum
 * $4$ Problems $6$


 * 7 Continued Fractions
 * $1$ Basic properties
 * $2$ Best approximation
 * $3$ Pell's equation
 * $4$ A set of real numbers modulo $1$
 * $5$ Problems $7$


 * 8 Transcendental Numbers
 * $1$ Liouville's theorem and applications
 * $2$ The Hermite and Lindemann theorems
 * $3$ The Gelfond-Schneider theorem
 * $4$ Problems $8$


 * 9 Quadratic Forms
 * $1$ Equivalence of forms
 * $2$ Sums of three squares
 * $3$ Representation by binary forms
 * $4$ Algorithms for reduced forms
 * $5$ Problems $9$


 * 10 Genera and the Class Group
 * $1$ The genus of a form
 * $2$ Composition and the class group
 * $3$ A formula for the class number
 * $4$ Problems $10$


 * 11 Partitions
 * $1$ Elementary properties
 * $2$ Jacobi's identity
 * $3$ Estimates for $\map p n$
 * $4$ Problems $11$


 * 12 The Prime Numbers
 * $1$ The results of Chebyshev and Bertrand
 * $2$ Series involving primes
 * $3$ Riemann zeta function
 * $4$ Problems $12$


 * 13 Two Major Theorems on the Primes
 * $1$ Dirichlet's theorem
 * $2$ PNT: preliminaries and Selberg's theorem
 * $3$ PNT: the main proof
 * $4$ Problems $13$


 * 14 Diophantine Equations
 * $1$ Legendre's theorem
 * $2$ Fermat's last theorem
 * $3$ Skolem's method
 * $4$ Mordell's equation
 * $5$ Problems $14$


 * 15 Elliptic Curves: Basic Theory
 * $1$ Geometric preliminaries
 * $2$ Rational points on elliptic curves
 * $3$ Mordell-Weil theorem
 * $4$ Problems $15$


 * 16 Elliptic Curves: Further Results and Applications
 * $1$ Weierstrass equation
 * $2$ Nagell-Lutz theorem
 * $3$ Curves defined over finite fields
 * $4$ Lenstra's factorization method
 * $5$ $L$-functions for curves
 * $6$ Problems $16$


 * Answers and Hints to Problems
 * Problems 1
 * Problems 2
 * Problems 3
 * Problems 4
 * Problems 5
 * Problems 6
 * Problems 7
 * Problems 8
 * Problems 9
 * Problems 10
 * Problems 11
 * Problems 12
 * Problems 13
 * Problems 14
 * Problems 15
 * Problems 16


 * Tables


 * Bibliography


 * Index of Notation


 * General Index



Source work progress
* : Preface to first edition: Prerequisites: $\text {(i)}$