Topics in number theory / Jorge Morales.

Format:

Book

Author/Creator:

Morales, Jorge, author.

Series:

London Mathematical Society lecture note series. ; 501.

London Mathematical Society lecture note series ; 501

Language:

English

Subjects (All):

Number theory.

Physical Description:

1 online resource (xiv, 370 pages) : illustrations (black and white), digital, PDF file(s).

Edition:

1st ed.

Place of Publication:

Cambridge : Cambridge University Press, 2026.

Summary:

Spanning elementary, algebraic, and analytic approaches, this book provides an introductory overview of essential themes in number theory. Designed for mathematics students, it progresses from undergraduate-accessible material requiring only basic abstract algebra to graduate-level topics demanding familiarity with algebra and complex analysis.

Contents:

Cover

Series page

Title page

Copyright page

Dedication

Epigraph

Contents

Preface

Part I Elementary Methods

1 Congruences and Primes

1.1 Euclidean Division

1.1.1 The Euclidean Algorithm for the GCD

1.1.2 The Complexity of the Euclidean Algorithm for the GCD

1.1.3 The Diophantine Equation ax + by + c = 0

1.2 Primes and Unique Factorization

1.2.1 The Fundamental Theorem of Arithmetic

1.2.2 A First Glance at the Riemann Zeta Function

1.2.3 The Series Σ 1/p

1.3 Congruences

1.3.1 The Ring Z/nZ

1.3.2 The Chinese Remainder Theorem

1.4 Arithmetic Functions

1.4.1 The Functions τ and σ

1.4.2 The Möbius Inversion Formula

1.4.3 Partitions

1.5 The Law of Quadratic Reciprocity

1.6 Some Applications

1.6.1 The Miller-Rabin Primality Test

1.6.2 RSA Encryption

2 Continued Fractions

2.1 Preliminaries on Möbius Transformations

2.2 Continued Fractions

2.2.1 RCF Representation of Rational Numbers

2.2.2 RCF Representation of Irrational Numbers

2.2.3 Approximation by Rational Numbers

2.2.4 Action of GL(2,Z) and SL(2,Z)

2.3 Quadratic Irrationalities

2.3.1 Periodicity of Quadratic Irrationalities

2.3.2 The Action of GL(2,Z) on Quadratic Irrationalities

2.4 Units of the Ring Z[√n]

2.5 Some Applications of Continued Fractions

2.5.1 The Pell Equation

2.5.2 The Negative Pell Equation

2.5.3 Sums of Two Squares

2.5.4 The Equation X[sup(2)] + dY[sup(2)] = p

2.5.5 The Wiener Attack on RSA

3 Euclidean and Principal Ideal Domains

3.1 Euclidean Domains

3.2 Principal Ideal Domains

3.3 Unique Factorization Domains

3.4 Modules over Principal Ideal Domains

3.4.1 Uniqueness of the Invariant Factors

3.4.2 Smith Normal Form

3.4.3 p-Decomposition of Torsion Modules

3.4.4 Finitely Generated Abelian Groups.

3.4.5 Jordan Canonical Form

3.4.6 Discrete Subgroups of R[sup(n)]

Part II Algebraic Methods

4 Some Commutative Algebra

4.1 The General Chinese Remainder Theorem

4.2 Localization

4.3 Noetherian Rings and Modules

4.3.1 Primary Decomposition

4.3.2 Krull Dimension

4.4 Fractional Ideals and Picard Groups

4.5 Finite-Dimensional Commutative Algebras

4.5.1 Separable Algebras

4.5.2 Traces, Norms, and Discriminants

5 Integrality

5.1 Integral Closure

5.2 Dedekind Domains

5.2.1 Fractional Ideals in Dedekind Domains

5.2.2 Fractional Ideals in One-Dimensional Noetherian Domains

5.2.3 Valuations

5.3 Modules over Dedekind Domains

5.4 Lattices and Symmetric Bilinear Forms

5.5 Ramification

5.5.1 Explicit Factorization of Primes

5.5.2 The Discriminant

5.5.3 The Norm of an Ideal

5.5.4 The Different

5.5.5 Integral Bases: A Theorem of Artin

6 Ideal Class Groups and Units

6.1 Preliminaries about Lattices in R[sup(n)]

6.1.1 Minkowski's Convex Body Theorem

6.1.2 Sums of Two and Four Squares

6.1.3 Sphere Packings

6.2 Embedding Number Fields in Euclidean Space

6.2.1 Finiteness of the Class Group

6.2.2 Volume Computations and Explicit Bounds

6.3 Dirichlet's Theorem on Units

6.4 The Analytic Class Number Formula

7 Quadratic Fields and Binary QuadraticForms

7.1 Ramification in Quadratic Number Fields

7.2 Units

7.3 Class Groups, Class Numbers, and Binary Quadratic Forms

7.3.1 Binary Quadratic Forms

7.3.2 Quadratic Orders

7.3.3 Binary Quadratic Forms on Ideals

7.3.4 The Narrow Picard Group and Binary Quadratic Forms

7.4 Reduction of Positive-Definite Binary Quadratic Forms

7.4.1 Definite-Positive Binary Quadratic Forms

7.5 Reduction of Indefinite Binary Quadratic Forms

7.5.1 Negative Regular Continued Fractions.

7.5.2 Representation of Rational Numbers

7.5.3 Representation of Irrational Numbers

7.5.4 The Action of SL(2,Z)on P[sup(1)](R) Revisited

7.5.5 Quadratic Irrationalities and NRCFs

7.5.6 Reduction of Binary Indefinite Quadratic Forms

8 Cyclotomic Fields

8.1 The Ring of Cyclotomic Integers

8.2 Second Proof of the Law of Quadratic Reciprocity

8.3 The Embedding of Quadratic Fields into Cyclotomic Fields

8.4 The First Case of Fermat's Last Theorem

Part III Analytic Methods

9 Dirichlet Series

9.1 Dirichlet Series and Laplace Transforms

9.2 L-Series

9.3 Primes in Arithmetic Progressions

9.4 Dedekind Zeta Functions of Quadratic Number Fields

9.5 Gauss Sums

9.5.1 The Computation of L(1,χ)

9.5.2 Class Number Formulas for Quadratic Number Fields

9.5.3 The Sign of the Quadratic Gauss Sum

10 The Riemann Zeta Function

10.1 Basic Properties

10.2 The Functional Equation

10.2.1 The Gamma Function

10.2.2 The Jacobi Theta Function

10.2.3 Proof of the Functional Equation

11 The Prime Number Theorem

11.1 A Brief History

11.2 Elementary Bounds

11.3 Proof of the Prime Number Theorem

11.4 The von Mangoldt Formula

11.5 Remainder Estimates

11.6 Riemann's Explicit Formula

References

Index.

Notes:

Includes bibliographical references and index.

Description based on online resource; title from PDF title page (viewed on April 9, 2026).

ISBN:

1-009-76047-5

1-009-76050-5

OCLC:

1581572328