Fine B., Rosenberger G. Number Theory: An Introduction via the Density of Primes

Second Edition. — Boston - Basel - Berlin: Birkhäuser, 2016 — 422 p. — ISBN: 978-3-319-43875-7 (eBook).A solid introduction to analytic number theory, including full proofs of Dirichlet's Theorem and the Prime Number Theorem.
Concise treatment of algebraic number theory, including a complete presentation of primes, prime factorizations in algebraic number fields, and unique factorization of ideals.
One of the few books to include the AKS algorithm that shows that primality testing is one of polynomial time. Many interesting ancillary topics, such as primality testing and cryptography, Fermat and Mersenne numbers, and Carmichael numbers.
Now in its second edition, this textbook provides an introduction and overview of number theory based on the density and properties of the prime numbers. This unique approach offers both a firm background in the standard material of number theory, as well as an overview of the entire discipline. All of the essential topics are covered, such as the fundamental theorem of arithmetic, theory of congruences, quadratic reciprocity, arithmetic functions, and the distribution of primes. New in this edition are coverage of p-adic numbers, Hensel's lemma, multiple zeta-values, and elliptic curve methods in primality testing.Introduction and Historical RemarksBasic Number Theory
The Ring of Integers
Divisibility, Primes, and Composites
The Fundamental Theorem of Arithmetic
Congruences and Modular Arithmetic
Basic Theory of Congruences
The Ring of Integers Mod N
Units and the Euler Phi Function
Fermat’s Little Theorem and the Order of an Element
On Cyclic Groups
The Solution of Polynomial Congruences Modulo m
Linear Congruences and the Chinese Remainder Theorem
Higher Degree Congruences
Quadratic Reciprocity
ExercisesThe Infinitude of Primes
The Infinitude of Primes
Some Direct Proofs and Variations
Some Analytic Proofs and Variations
The Fermat and Mersenne Numbers
The Fibonacci Numbers and the Golden Section
Some Simple Cases of Dirichlet’s Theorem
A Topological Proof and a Proof Using Codes
Sums of Squares
Pythagorean Triples
Fermat’s Two-Square Theorem
The Modular Group
Lagrange’s Four Square Theorem
The Infinitude of Primes Through Continued Fractions
Dirichlet’s Theorem
Twin Prime Conjecture and Related Ideas
Primes Between x and 2x
Arithmetic Functions and the Mö bius Inversion Formula
ExercisesThe Density of Primes
The Prime Number Theorem — Estimates and History
Chebyshev’s Estimate and Some Consequences
Equivalent Formulations of the Prime Number Theorem
The Riemann Zeta Function and the Riemann Hypothesis
The Real Zeta Function of Euler
Analytic Functions and Analytic Continuation
The Riemann Zeta Function
The Prime Number Theorem
The Elementary Proof
Multiple Zeta Values
Some Extensions and Comments
ExercisesPrimality Testing — An Overview
Primality Testing and Factorization
Sieving Methods
Brun’s Sieve and Brun’s Theorem
Primality Testing and Prime Records
Pseudo-Primes and Probabilistic Testing
The Lucas–Lehmer Test and Prime Records
Some Additional Primality Tests
Elliptic Curve Methods
Cryptography and Primes
Some Number Theoretic Cryptosystems
Public Key Cryptography and the RSA Algorithm
Elliptic Curve Cryptography
The AKS Algorithm
ExercisesPrimes and Algebraic Number Theory
Algebraic Number Theory
Unique Factorization Domains
Euclidean Domains and the Gaussian Integers
Principal Ideal Domains
Prime and Maximal Ideals
Algebraic Number Fields
Algebraic Extensions of Q
Algebraic and Transcendental Numbers
Symmetric Polynomials
Discriminant and Norm
Algebraic Integers
The Ring of Algebraic Integers
Integral Bases
Quadratic Fields and Quadratic Integers
The Transcendence of e and π
The Geometry of Numbers — Minkowski Theory
Dirichlet’s Unit Theorem
The Theory of Ideals
Unique Factorization of Ideals
An Application of Unique Factorization
The Ideal Class Group
Norms of Ideals
Class Number
ExercisesThe Fields [b]Q_p of p-Adic Numbers: Hensel’s Lemma[/b]
The p-Adic Fields and p-Adic Expansions
The Construction of the Real Numbers
The Completeness of Real Numbers
The Construction of R
The Characterization of R
Normed Fields and Cauchy Completions
The p-Adic Fields
The p-Adic Norm
The Construction of Q_p
p-Adic Arithmetic and p-Adic Expansions
The p-Adic Integers
Principal Ideals and Unique Factorization
The Completeness of Z_p
Ostrowski’s Theorem
Hensel’s Lemma and Applications
The Non-isomorphism of the p-Adic Fields
Exercises