Cover -- Half-title -- Title -- Copyright -- Dedication -- Contents -- List of Tables -- Notation -- Introduction -- 1 Integral Domains -- 1.1 Integral Domains -- Properties of an Integral Domain -- Properties of Divisors -- Properties of Units -- Properties of Associates -- 1.2 Irreducibles and Primes -- 1.3 Ideals -- 1.4 Principal Ideal Domains -- 1.5 Maximal Ideals and Prime Ideals -- 1.6 Sums and Products of Ideals -- Exercises -- Suggested Reading -- Biographies -- 2 Euclidean Domains -- 2.1 Euclidean Domains -- 2.2 Examples of Euclidean Domains -- 2.3 Examples of Domains That are Not Euclidean -- 2.4 Almost Euclidean Domains -- 2.5 Representing Primes by Binary Quadratic Forms -- Exercises -- Suggested Reading -- Biographies -- 3 Noetherian Domains -- 3.1 Noetherian Domains -- 3.2 Factorization Domains -- 3.3 Unique Factorization Domains -- 3.4 Modules -- 3.5 Noetherian Modules -- Exercises -- Suggested Reading -- Biographies -- 4 Elements Integral over a Domain -- 4.1 Elements Integral over a Domain -- 4.2 Integral Closure -- Exercises -- Suggested Reading -- Biographies -- 5 Algebraic Extensions of a Field -- 5.1 Minimal Polynomial of an Element Algebraic over a Field -- 5.2 Conjugates of alpha over K -- 5.3 Conjugates of an Algebraic Integer -- 5.4 Algebraic Integers in a Quadratic Field -- 5.5 Simple Extensions -- 5.6 Multiple Extensions -- Exercises -- Suggested Reading -- Biographies -- 6 Algebraic Number Fields -- 6.1 Algebraic Number Fields -- 6.2 Conjugate Fields of an Algebraic Number Field -- 6.3 The Field Polynomial of an Element of an Algebraic Number Field -- 6.4 The Discriminant of a Set of Elements in an Algebraic Number Field -- 6.5 Basis of an Ideal -- 6.6 Prime Ideals in Rings of Integers -- Exercises -- Suggested Reading -- Biographies -- 7 Integral Bases -- 7.1 Integral Basis of an Algebraic Number Field.

7.2 Minimal Integers -- 7.3 Some Integral Bases in Cubic Fields -- 7.4 Index and Minimal Index of an Algebraic Number Field -- 7.5 Integral Basis of a Cyclotomic Field -- Exercises -- Suggested Reading -- Biographies -- 8 Dedekind Domains -- 8.1 Dedekind Domains -- 8.2 Ideals in a Dedekind Domain -- 8.3 Factorization into Prime Ideals -- 8.4 Order of an Ideal with Respect to a Prime Ideal -- 8.5 Generators of Ideals in a Dedekind Domain -- Exercises -- Suggested Reading -- 9 Norms of Ideals -- 9.1 Norm of an Integral Ideal -- 9.2 Norm and Trace of an Element -- 9.3 Norm of a Product of Ideals -- 9.4 Norm of a Fractional Ideal -- Exercises -- Suggested Reading -- Biographies -- 10 Factoring Primes in a Number Field -- 10.1 Norm of a Prime Ideal -- 10.2 Factoring Primes in a Quadratic Field -- 10.3 Factoring Primes in a Monogenic Number Field -- 10.4 Some Factorizations in Cubic Fields -- 10.5 Factoring Primes in an Arbitrary Number Field -- 10.6 Factoring Primes in a Cyclotomic Field -- Exercises -- Suggested Reading -- 11 Units in Real Quadratic Fields -- 11.1 The Units of… -- 11.2 The Equation… -- 11.3 Units of Norm 1 -- 11.4 Units of Norm -1 -- 11.5 The Fundamental Unit -- 11.6 Calculating the Fundamental Unit -- 11.7 The Equation… -- Exercises -- Suggested Reading -- Biographies -- 12 The Ideal Class Group -- 12.1 Ideal Class Group -- 12.2 Minkowski's Translate Theorem -- 12.3 Minkowski's Convex Body Theorem -- 12.4 Minkowski's Linear Forms Theorem -- 12.5 Finiteness of the Ideal Class Group -- 12.6 Algorithm to Determine the Ideal Class Group -- 12.7 Applications to Binary Quadratic Forms -- Exercises -- Suggested Reading -- Biographies -- 13 Dirichlet's Unit Theorem -- 13.1 Valuations of an Element of a Number Field -- 13.2 Properties of Valuations -- 13.3 Proof of Dirichlet's Unit Theorem -- 13.4 Fundamental System of Units.

13.5 Roots of Unity -- 13.6 Fundamental Units in Cubic Fields -- 13.7 Regulator -- Exercises -- Suggested Reading -- Biographies -- 14 Applications to Diophantine Equations -- 14.1 Insolvability of…Using Congruence Considerations -- 14.2 Solving…Using Algebraic Numbers -- 14.3 The Diophantine Equation -- Exercises -- Suggested Reading -- Biographies -- List of Definitions -- Location of Theorems -- Location of Lemmas -- Bibliography -- Index.