# Introduction To Number Theory

## Hardcover

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## Overview

Introduction to Number Theory is dedicated to concrete questions about integers, to place an emphasis on problem solving by students. When undertaking a first course in number theory, students enjoy actively engaging with the properties and relationships of numbers.The book begins with introductory material, including uniqueness of factorization of integers and polynomials. Subsequent topics explore quadratic reciprocity, Hensel's Lemma, p-adic powers series such as exp(px) and log(1+px), the Euclidean property of some quadratic rings, representation of integers as norms from quadratic rings, and Pell's equation via continued fractions.Throughout the five chapters and more than 100 exercises and solutions, readers gain the advantage of a number theory book that focuses on doing calculations. This textbook is a valuable resource for undergraduates or those with a background in university level mathematics.

## Product Details

ISBN-13: 9781786344717 World Scientific Publishing Europe Ltd 01/30/2018 Essential Textbooks In Mathematics Series 264 6.00(w) x 9.00(h) x 0.63(d)

About the Author v

Acknowledgments vii

Introduction xi

1 Euclid's Algorithm 1

1.1 Some Examples of Rings 1

1.2 Euclid's Algorithm 5

1.3 Invertible Elements Modulo n 9

1.4 Solving Linear Congruences 13

1.5 The Chinese Remainder Theorem 16

1.6 Prime Numbers 19

Hints for Some Exercises 26

2 Polynomial Rings 27

2.1 Long Division of Polynomials 28

2.2 Highest Common Factors 34

2.3 Uniqueness of Factorization 38

2.4 Irreducible Polynomials 40

2.5 Unique Factorization Domains 46

Hints for Some Exercises 50

3 Congruences Modulo Prime Numbers 53

3.1 Fermat's Little Theorem 55

3.2 The Euler Totient Function 57

3.3 Cyclotomic Polynomials and Primitive Roots 61

3.4 Public Key Cryptography 68

3.5 Quadratic Reciprocity 73

3.6 Congruences in an Arbitrary Ring 84

3.7 Proof of the Second Nebensatz 91

3.8 Gauss Sums and the Proof of Quadratic Reciprocity 93

Hints for Some Exercises 99

4 p-Adic Methods in Number Theory 101

4.1 Hensel's Lemma 101

4.2 Quadratic Congruences 112

4.3 p-Adic Convergence of Series 116

4.4 p-Adic Logarithms and Exponential Maps 124

4.5 Teichmüller Lifts 130

4.6 The Ring of p-Adic Integers 139

Hints for Some Exercises 148

5 Diophantine Equations and Quadratic Rings 149

5.1 Diophantine Equations and Unique Factorization 149

5.2 Quadratic Rings 152

5.3 Norm-Euclidean Quadratic Rings 159

5.4 Decomposing Primes in Quadratic Rings 167

5.5 Continued Fractions 176

5.6 Pell's Equation 185

5.7 Real Quadratic Rings and Diophantine Equations 193

Hints for Some Exercises 199

Solution to Exercises 201

Bibliography 243

Index 245

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