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Bringing the material up to date to reflect modern applications, **Algebraic Number Theory, Second Edition** has been completely rewritten and reorganized to incorporate a new style, methodology, and presentation. This edition focuses on integral domains, ideals, and unique factorization in the first chapter; field extensions in the second chapter; and class groups in the third chapter. Applications are now collected in chapter four and at the end of chapter five, where primality testing is highlighted as an application of the Kronecker–Weber theorem. In chapter five, the sections on ideal decomposition in number fields have been more evenly distributed. The final chapter continues to cover reciprocity laws.

**New to the Second Edition**

- Reorganization of all chapters
- More complete and involved treatment of Galois theory
- A study of binary quadratic forms and a comparison of the ideal and form class groups
- More comprehensive section on Pollard’s cubic factoring algorithm
- More detailed explanations of proofs, with less reliance on exercises, to provide a sound understanding of challenging material

The book includes mini-biographies of notable mathematicians, convenient cross-referencing, a comprehensive index, and numerous exercises. The appendices present an overview of all the concepts used in the main text, an overview of sequences and series, the Greek alphabet with English transliteration, and a table of Latin phrases and their English equivalents.

Suitable for a one-semester course, this accessible, self-contained text offers broad, in-depth coverage of numerous applications. Readers are lead at a measured pace through the topics to enable a clear understanding of the pinnacles of algebraic number theory.

**Integral Domains, Ideals, and Unique Factorization**

Integral Domains

Factorization Domains

Ideals

Noetherian and Principal Ideal Domains

Dedekind Domains

Algebraic Numbers and Number Fields

Quadratic Fields

**Field Extensions **

Automorphisms, Fixed Points, and Galois Groups

Norms and Traces

Integral Bases and Discriminants

Norms of Ideals

**Class Groups**

Binary Quadratic Forms

Forms and Ideals

Geometry of Numbers and the Ideal Class Group

Units in Number Rings

Dirichlet’s Unit Theorem

**Applications: Equations and Sieves **

Prime Power Representation

Bachet’s Equation

The Fermat Equation

Factoring

The Number Field Sieve

**Ideal Decomposition in Number Fields**

Inertia, Ramification, and Splitting of Prime Ideals

The Different and Discriminant

Ramification

Galois Theory and Decomposition

Kummer Extensions and Class-Field Theory

The Kronecker–Weber Theorem

An Application—Primality Testing

**Reciprocity Laws**

Cubic Reciprocity

The Biquadratic Reciprocity Law

The Stickelberger Relation

The Eisenstein Reciprocity Law

**Appendix A: Abstract Algebra**

**Appendix B: Sequences and Series**

**Appendix C: The Greek Alphabet**

**Appendix D: Latin Phrases**

Bibliography

**Solutions to Odd-Numbered Exercises **

**Index**

Bringing the material up to date to reflect modern applications, **Algebraic Number Theory, Second Edition** has been completely rewritten and reorganized to incorporate a new style, methodology, and presentation. This edition focuses on integral domains, ideals, and unique factorization in the first chapter; field extensions in the second chapter; and class groups in the third chapter. Applications are now collected in chapter four and at the end of chapter five, where primality testing is highlighted as an application of the Kronecker–Weber theorem. In chapter five, the sections on ideal decomposition in number fields have been more evenly distributed. The final chapter continues to cover reciprocity laws.

**New to the Second Edition**

- Reorganization of all chapters
- More complete and involved treatment of Galois theory
- A study of binary quadratic forms and a comparison of the ideal and form class groups
- More comprehensive section on Pollard’s cubic factoring algorithm
- More detailed explanations of proofs, with less reliance on exercises, to provide a sound understanding of challenging material

The book includes mini-biographies of notable mathematicians, convenient cross-referencing, a comprehensive index, and numerous exercises. The appendices present an overview of all the concepts used in the main text, an overview of sequences and series, the Greek alphabet with English transliteration, and a table of Latin phrases and their English equivalents.

Suitable for a one-semester course, this accessible, self-contained text offers broad, in-depth coverage of numerous applications. Readers are lead at a measured pace through the topics to enable a clear understanding of the pinnacles of algebraic number theory.

**Integral Domains, Ideals, and Unique Factorization**

Integral Domains

Factorization Domains

Ideals

Noetherian and Principal Ideal Domains

Dedekind Domains

Algebraic Numbers and Number Fields

Quadratic Fields

**Field Extensions **

Automorphisms, Fixed Points, and Galois Groups

Norms and Traces

Integral Bases and Discriminants

Norms of Ideals

**Class Groups**

Binary Quadratic Forms

Forms and Ideals

Geometry of Numbers and the Ideal Class Group

Units in Number Rings

Dirichlet’s Unit Theorem

**Applications: Equations and Sieves **

Prime Power Representation

Bachet’s Equation

The Fermat Equation

Factoring

The Number Field Sieve

**Ideal Decomposition in Number Fields**

Inertia, Ramification, and Splitting of Prime Ideals

The Different and Discriminant

Ramification

Galois Theory and Decomposition

Kummer Extensions and Class-Field Theory

The Kronecker–Weber Theorem

An Application—Primality Testing

**Reciprocity Laws**

Cubic Reciprocity

The Biquadratic Reciprocity Law

The Stickelberger Relation

The Eisenstein Reciprocity Law

**Appendix A: Abstract Algebra**

**Appendix B: Sequences and Series**

**Appendix C: The Greek Alphabet**

**Appendix D: Latin Phrases**

Bibliography

**Solutions to Odd-Numbered Exercises **

**Index**

Bringing the material up to date to reflect modern applications, **Algebraic Number Theory, Second Edition** has been completely rewritten and reorganized to incorporate a new style, methodology, and presentation. This edition focuses on integral domains, ideals, and unique factorization in the first chapter; field extensions in the second chapter; and class groups in the third chapter. Applications are now collected in chapter four and at the end of chapter five, where primality testing is highlighted as an application of the Kronecker–Weber theorem. In chapter five, the sections on ideal decomposition in number fields have been more evenly distributed. The final chapter continues to cover reciprocity laws.

**New to the Second Edition**

- Reorganization of all chapters
- More complete and involved treatment of Galois theory
- A study of binary quadratic forms and a comparison of the ideal and form class groups
- More comprehensive section on Pollard’s cubic factoring algorithm
- More detailed explanations of proofs, with less reliance on exercises, to provide a sound understanding of challenging material

The book includes mini-biographies of notable mathematicians, convenient cross-referencing, a comprehensive index, and numerous exercises. The appendices present an overview of all the concepts used in the main text, an overview of sequences and series, the Greek alphabet with English transliteration, and a table of Latin phrases and their English equivalents.

Suitable for a one-semester course, this accessible, self-contained text offers broad, in-depth coverage of numerous applications. Readers are lead at a measured pace through the topics to enable a clear understanding of the pinnacles of algebraic number theory.

**Integral Domains, Ideals, and Unique Factorization**

Integral Domains

Factorization Domains

Ideals

Noetherian and Principal Ideal Domains

Dedekind Domains

Algebraic Numbers and Number Fields

Quadratic Fields

**Field Extensions **

Automorphisms, Fixed Points, and Galois Groups

Norms and Traces

Integral Bases and Discriminants

Norms of Ideals

**Class Groups**

Binary Quadratic Forms

Forms and Ideals

Geometry of Numbers and the Ideal Class Group

Units in Number Rings

Dirichlet’s Unit Theorem

**Applications: Equations and Sieves **

Prime Power Representation

Bachet’s Equation

The Fermat Equation

Factoring

The Number Field Sieve

**Ideal Decomposition in Number Fields**

Inertia, Ramification, and Splitting of Prime Ideals

The Different and Discriminant

Ramification

Galois Theory and Decomposition

Kummer Extensions and Class-Field Theory

The Kronecker–Weber Theorem

An Application—Primality Testing

**Reciprocity Laws**

Cubic Reciprocity

The Biquadratic Reciprocity Law

The Stickelberger Relation

The Eisenstein Reciprocity Law

**Appendix A: Abstract Algebra**

**Appendix B: Sequences and Series**

**Appendix C: The Greek Alphabet**

**Appendix D: Latin Phrases**

Bibliography

**Solutions to Odd-Numbered Exercises **

**Index**

**Algebraic Number Theory, Second Edition** has been completely rewritten and reorganized to incorporate a new style, methodology, and presentation. This edition focuses on integral domains, ideals, and unique factorization in the first chapter; field extensions in the second chapter; and class groups in the third chapter. Applications are now collected in chapter four and at the end of chapter five, where primality testing is highlighted as an application of the Kronecker–Weber theorem. In chapter five, the sections on ideal decomposition in number fields have been more evenly distributed. The final chapter continues to cover reciprocity laws.

**New to the Second Edition**

- Reorganization of all chapters
- More complete and involved treatment of Galois theory
- A study of binary quadratic forms and a comparison of the ideal and form class groups
- More comprehensive section on Pollard’s cubic factoring algorithm

**Integral Domains, Ideals, and Unique Factorization**

Integral Domains

Factorization Domains

Ideals

Noetherian and Principal Ideal Domains

Dedekind Domains

Algebraic Numbers and Number Fields

Quadratic Fields

**Field Extensions **

Automorphisms, Fixed Points, and Galois Groups

Norms and Traces

Integral Bases and Discriminants

Norms of Ideals

**Class Groups**

Binary Quadratic Forms

Forms and Ideals

Geometry of Numbers and the Ideal Class Group

Units in Number Rings

Dirichlet’s Unit Theorem

**Applications: Equations and Sieves **

Prime Power Representation

Bachet’s Equation

The Fermat Equation

Factoring

The Number Field Sieve

**Ideal Decomposition in Number Fields**

Inertia, Ramification, and Splitting of Prime Ideals

The Different and Discriminant

Ramification

Galois Theory and Decomposition

Kummer Extensions and Class-Field Theory

The Kronecker–Weber Theorem

An Application—Primality Testing

**Reciprocity Laws**

Cubic Reciprocity

The Biquadratic Reciprocity Law

The Stickelberger Relation

The Eisenstein Reciprocity Law

**Appendix A: Abstract Algebra**

**Appendix B: Sequences and Series**

**Appendix C: The Greek Alphabet**

**Appendix D: Latin Phrases**

Bibliography

**Solutions to Odd-Numbered Exercises **

**Index**

**Algebraic Number Theory, Second Edition** has been completely rewritten and reorganized to incorporate a new style, methodology, and presentation. This edition focuses on integral domains, ideals, and unique factorization in the first chapter; field extensions in the second chapter; and class groups in the third chapter. Applications are now collected in chapter four and at the end of chapter five, where primality testing is highlighted as an application of the Kronecker–Weber theorem. In chapter five, the sections on ideal decomposition in number fields have been more evenly distributed. The final chapter continues to cover reciprocity laws.

**New to the Second Edition**

- Reorganization of all chapters
- More complete and involved treatment of Galois theory
- A study of binary quadratic forms and a comparison of the ideal and form class groups
- More comprehensive section on Pollard’s cubic factoring algorithm

**Integral Domains, Ideals, and Unique Factorization**

Integral Domains

Factorization Domains

Ideals

Noetherian and Principal Ideal Domains

Dedekind Domains

Algebraic Numbers and Number Fields

Quadratic Fields

**Field Extensions **

Automorphisms, Fixed Points, and Galois Groups

Norms and Traces

Integral Bases and Discriminants

Norms of Ideals

**Class Groups**

Binary Quadratic Forms

Forms and Ideals

Geometry of Numbers and the Ideal Class Group

Units in Number Rings

Dirichlet’s Unit Theorem

**Applications: Equations and Sieves **

Prime Power Representation

Bachet’s Equation

The Fermat Equation

Factoring

The Number Field Sieve

**Ideal Decomposition in Number Fields**

Inertia, Ramification, and Splitting of Prime Ideals

The Different and Discriminant

Ramification

Galois Theory and Decomposition

Kummer Extensions and Class-Field Theory

The Kronecker–Weber Theorem

An Application—Primality Testing

**Reciprocity Laws**

Cubic Reciprocity

The Biquadratic Reciprocity Law

The Stickelberger Relation

The Eisenstein Reciprocity Law

**Appendix A: Abstract Algebra**

**Appendix B: Sequences and Series**

**Appendix C: The Greek Alphabet**

**Appendix D: Latin Phrases**

Bibliography

**Solutions to Odd-Numbered Exercises **

**Index**

**Algebraic Number Theory, Second Edition** has been completely rewritten and reorganized to incorporate a new style, methodology, and presentation. This edition focuses on integral domains, ideals, and unique factorization in the first chapter; field extensions in the second chapter; and class groups in the third chapter. Applications are now collected in chapter four and at the end of chapter five, where primality testing is highlighted as an application of the Kronecker–Weber theorem. In chapter five, the sections on ideal decomposition in number fields have been more evenly distributed. The final chapter continues to cover reciprocity laws.

**New to the Second Edition**

- Reorganization of all chapters
- More complete and involved treatment of Galois theory
- A study of binary quadratic forms and a comparison of the ideal and form class groups
- More comprehensive section on Pollard’s cubic factoring algorithm

**Integral Domains, Ideals, and Unique Factorization**

Integral Domains

Factorization Domains

Ideals

Noetherian and Principal Ideal Domains

Dedekind Domains

Algebraic Numbers and Number Fields

Quadratic Fields

**Field Extensions **

Automorphisms, Fixed Points, and Galois Groups

Norms and Traces

Integral Bases and Discriminants

Norms of Ideals

**Class Groups**

Binary Quadratic Forms

Forms and Ideals

Geometry of Numbers and the Ideal Class Group

Units in Number Rings

Dirichlet’s Unit Theorem

**Applications: Equations and Sieves **

Prime Power Representation

Bachet’s Equation

The Fermat Equation

Factoring

The Number Field Sieve

**Ideal Decomposition in Number Fields**

Inertia, Ramification, and Splitting of Prime Ideals

The Different and Discriminant

Ramification

Galois Theory and Decomposition

Kummer Extensions and Class-Field Theory

The Kronecker–Weber Theorem

An Application—Primality Testing

**Reciprocity Laws**

Cubic Reciprocity

The Biquadratic Reciprocity Law

The Stickelberger Relation

The Eisenstein Reciprocity Law

**Appendix A: Abstract Algebra**

**Appendix B: Sequences and Series**

**Appendix C: The Greek Alphabet**

**Appendix D: Latin Phrases**

Bibliography

**Solutions to Odd-Numbered Exercises **

**Index**