This book is intended to complement my Elements of Algebra, and it is similarly motivated by the problem of solving polynomial equations.However, it is independent of the algebra book, and probably easier. In Elements of Algebra we sought solution by radicals, and this led to theconcepts of fields and groups and their fusion in the celebrated theory of Galois. In the present book we seek integer solutions, and this leads to the concepts of rings and ideals which merge in the equally celebrated theo of ideals due to Kummer and Dedekind.

Preface

1 Natural numbers and integers

 1.1 Natural numbers

 1.2 Induction

 1.3 Integers

 1.4 Division with remainder

 1.5 Binary notation

 1.6 Diophantine equations

 1.7 TheDiophantus chord method

 1.8 Gaussian integers

 1.9 Discussion

2 The Euclidean algorithm

 2.1 The gcd by subtraction

 2.2 The gcd by division with remainder

 2.3 Linear representation of the gcd

 2.4 Primes and factorization

 2.5 Consequences of unique prime factorization

 2.6 Linear Diophantine equations

 2.7 The vector Euclidean algorithm

 2.8 The map of relatively prime pairs

 2.9 Discussion

3 Congruence arithmetic

 3.1 Congruence mod n

 3.2 Congruence classes and their arithmetic

 3.3 Inverses modp

3.4 Fermat's little theorem

 3.5 Congruence theorems of Wilson and Lagrange

 3.6 Inverses mod k

 3.7 Quadratic Diophantine equations

 3.8 Primitive roots

 3.9 Existence of primitive roots

 3.10 Discussion

4 The RSA eryptosystem

 4.1 Trapdoor functions

 4.2 Ingredients of RSA

 4.3 Exponentiation mod n

 4.4 RSA encryption and decryption

 4.5 Digital signatures

 4.6 Other computational issues

 4.7 Discussion

5 The Pell equation

 5.1 Side and diagonal numbers

 5.2 The equation x2-2y2 = 1

 5.3 The group of solutions

 5.4 The general Pell equation and Z[n]

 5.5 The pigeonhole argument

 5.6 Quadratic forms

 5.7 The map of primitive vectors

 5.8 Periodicity in the map ofx2 -ny2

 5.9 Discussion

6 The Gaussian integers

 6.1 Zand its norm

 6.2 Divisibility and primes in Zand Z

 6.3 Conjugates

 6.4 Division in Z[i]

 6.5 Fermat's two square theorem

 6.6 Pythagorean triples

 6.7 Primes of the form 4n+1

 6.8 Discussion

7 Quadratic integers

 7.1 The equation y3=x2+2

 7.2 The division property in Z[-2]

 7.3 The gcd in Z[-2]

 7.4 Z[-3] and Z[ζ3]

 7.5 Rational solutions of x3+y3=z3+w3

 7.6 The prime -3 in Z[ζ3]

 7.7 Fermat's last theorem for n=3

 7.8 Discussion

8 The four square theorem

 8.1 Real matrices and C

 8.2 Complex matrices and H

 8.3 The quaternion units

 8.4 Z[i,j,k]

 8.5 The Hurwitz integers

 8.6 Conjugates

 8.7 A prime divisor property

 8.8 Proof of the four square theorem

 8.9 Discussion

9 Quadratic reciprocity

 9.1 Primes x2+y2, x2+2y2, and x2+3y2

 9.2 Statement of quadratic reciprocity

 9.3 Euler's criterion

 9.4 The value of (2/q)

 9.5 The story so far

 9.6 The Chinese remainder theorem

 9.7 The full Chinese remainder theorem

 9.8 Proof of quadratic reciprocity

 9.9 Discussion

10 Rings

 10.1 The ring axioms

 10.2 Rings and fields

 10.3 Algebraic integers

 10.4 Quadratic fields and their integers

 10.5 Norm and units of quadratic fields

 10.6 Discussion

11 Ideals

 11.1 Ideals and the gcd

 11.2 Ideals and divisibility in Z

 11.3 Principal ideal domains

 11.4 A nonprincipal ideal of Z[-3]

 11.5 A nonprincipal ideal of Z[-5]

 11.6 Ideals of imaginary quadratic fields as lattices

 11.7 Products and prime ideals

 11.8 Ideal prime factorization

 11.9 Discussion

12 Prime ideals

 12.1 Ideals and congruence

 12.2 Prime and maximal ideals

 12.3 Prime ideals of imaginary quadratic fields

 12.4 Conjugate ideals

 12.5 Divisibility and containment

 12.6 Factorization of ideals

 12.7 Ideal classes

 12.8 Primes of the form X2+5y2

 12.9 Discussion

Bibliography

Index