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

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

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

ISBN 0-19-852376-9

Subject Matter

• Number Theory

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

Next

Further Editions

• 1988: H.E. Rose: A Course in Number Theory

Source work progress

• 1994: H.E. Rose: A Course in Number Theory (2nd ed.) ... (previous) ... (next): $1$ Divisibility: $1.1$ The Euclidean algorithm and unique factorization: Theorem $1.2$