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ISBN 10: 110701901X
ISBN 13: 9781107019010
Author: Alan Baker
Developed from the author's popular text, A Concise Introduction to the Theory of Numbers, this book provides a comprehensive initiation to all the major branches of number theory. Beginning with the rudiments of the subject, the author proceeds to more advanced topics, including elements of cryptography and primality testing, an account of number fields in the classical vein including properties of their units, ideals and ideal classes, aspects of analytic number theory including studies of the Riemann zeta-function, the prime-number theorem and primes in arithmetical progressions, a description of the Hardy–Littlewood and sieve methods from respectively additive and multiplicative number theory and an exposition of the arithmetic of elliptic curves. The book includes many worked examples, exercises and further reading. Its wider coverage and versatility make this book suitable for courses extending from the elementary to beginning graduate studies.
1 Divisibility
1.1 Foundations
1.2 Division algorithm
1.3 Greatest common divisor
1.4 Euclid's algorithm
1.5 Fundamental theorem
1.6 Properties of the primes
1.7 Further reading
1.8 Exercises
2 Arithmetical functions
2.1 The function [x]
2.2 Multiplicative functions
2.3 Euler's (totient) function ø(n)
2.4 The Möbius function μ(n)
2.5 The functions τ(n) and σ(n)
2.6 Average orders
2.7 Perfect numbers
2.8 The Riemann zeta-function
2.9 Further reading
2.10 Exercises
3 Congruences
3.1 Definitions
3.2 Chinese remainder theorem
3.3 The theorems of Fermat and Euler
3.4 Wilson's theorem
3.5 Lagrange's theorem
3.6 Primitive roots
3.7 Indices
3.8 Further reading
3.9 Exercises
4 Quadratic residues
4.1 Legendre's symbol
4.2 Euler's criterion
4.3 Gauss' lemma
4.4 Law of quadratic reciprocity
4.5 Jacobi's symbol
4.6 Further reading
4.7 Exercises
5 Quadratic forms
5.1 Equivalence
5.2 Reduction
5.3 Representations by binary forms
5.4 Sums of two squares
5.5 Sums of four squares
5.6 Further reading
5.7 Exercises
6 Diophantine approximation
6.1 Dirichlet's theorem
6.2 Continued fractions
6.3 Rational approximations
6.4 Quadratic irrationals
6.5 Liouville's theorem
6.6 Transcendental numbers
6.7 Minkowski's theorem
6.8 Further reading
6.9 Exercises
7 Quadratic fields
7.1 Algebraic number fields
7.2 The quadratic field
7.3 Units
7.4 Primes and factorization
7.5 Euclidean fields
7.6 The Gaussian field
7.7 Further reading
7.8 Exercises
8 Diophantine equations
8.1 The Pell equation
8.2 The Thue equation
8.3 The Mordell equation
8.4 The Fermat equation
8.5 The Catalan equation
8.6 The abc-conjecture
8.7 Further reading
8.8 Exercises
9 Factorization and primality testing
9.1 Fermat pseudoprimes
9.2 Euler pseudoprimes
9.3 Fermat factorization
9.4 Fermat bases
9.5 The continued-fraction method
9.6 Pollard's method
9.7 Cryptography
9.8 Further reading
9.9 Exercises
10 Number fields
10.1 Introduction
10.2 Algebraic numbers
10.3 Algebraic number fields
10.4 Dimension theorem
10.5 Norm and trace
10.6 Algebraic integers
10.7 Basis and discriminant
10.8 Calculation of bases
10.9 Further reading
10.10 Exercises
11 Ideals
11.1 Origins
11.2 Definitions
11.3 Principal ideals
11.4 Prime ideals
11.5 Norm of an ideal
11.6 Formula for the norm
11.7 The different
11.8 Further reading
11.9 Exercises
12 Units and ideal classes
12.1 Units
12.2 Dirichlet's unit theorem
12.3 Ideal classes
12.4 Minkowski's constant
12.5 Dedekind's theorem
12.6 The cyclotomic field
12.7 Calculation of class numbers
12.8 Local fields
12.9 Further reading
12.10 Exercises
13 Analytic number theory
13.1 Introduction
13.2 Dirichlet series
13.3 Tchebychev's estimates
13.4 Partial summation formula
13.5 Mertens' results
13.6 The Tchebychev functions
13.7 The irrationality of ζ(3)
13.8 Further reading
13.9 Exercises
14 On the zeros of the zeta-function
14.1 Introduction
14.2 The functional equation
14.3 The Euler product
14.4 On the logarithmic derivative of ζ(s)
14.5 The Riemann hypothesis
14.6 Explicit formula for ζ'(s)/ζ(s)
14.7 On certain sums
14.8 The Riemann–von Mangoldt formula
14.9 Further reading
14.10 Exercises
15 On the distribution of the primes
15.1 The prime-number theorem
15.2 Refinements and developments
15.3 Dirichlet characters
15.4 Dirichlet L-functions
15.5 Primes in arithmetical progressions
15.6 The class number formulae
15.7 Siegel's theorem
15.8 Further reading
15.9 Exercises
16 The sieve and circle methods
16.1 The Eratosthenes sieve
16.2 The Selberg upper-bound sieve
16.3 Applications of the Selberg sieve
16.4 The large sieve
16.5 The circle method
16.6 Additive prime number theory
16.7 Further reading
16.8 Exercises
17 Elliptic curves
17.1 Introduction
17.2 The Weierstrass -function
17.3 The Mordell–Weil group
17.4 Heights on elliptic curves
17.5 The Mordell–Weil theorem
17.6 Computing the torsion subgroup
17.7 Conjectures on the rank
17.8 Isogenies and endomorphisms
17.9 Further reading
17.10 Exercises
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