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

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H.E. Rose: A Course in Number Theory (2nd Edition)

Published $\text {1994}$, Oxford Science Publications

ISBN 0-19-852376-9.


Subject Matter


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


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