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 $1987$)
Acknowledgements
$1$ 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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