| Description |
1 online resource (xvii, 428 pages) |
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data file |
| Physical Medium |
polychrome |
| Bibliography |
Includes bibliographical references (pages 423-424) and index. |
| Contents |
Integral domains -- Euclidean domains -- Noetherian domains -- Elements integral over a domain -- Algebraic extensions of a field -- Algebraic number fields -- Integral bases -- Dedekind domains -- Norms of ideals -- Decomposing primes in a number field -- Units in real quadratic fields -- The ideal class group -- Dirichlet's unit theorem -- Applications to diophantine equations. |
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Machine generated contents note: 1.1 Integral Domains -- 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 -- 2.1 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 -- 3.1 Noetherian Domains -- 3.2 Factorization Domains -- 3.3 Unique Factorization Domains -- 3.4 Modules -- 3.5 Noetherian Modules -- Exercises -- Suggested Reading -- Biographies -- 4.1 Elements Integral over a Domain -- 4.2 Integral Closure -- Exercises -- Suggested Reading -- Biographies -- 5.1 Minimal Polynomial of an Element Algebraic over a Field -- 5.2 Conjugates of α over -- 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 -- 6.1 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 -- 7.1 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 -- 8.1 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 -- 9.1 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 -- 10.1 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 -- 11.1 The Units of Z+Z/2 -- 11.2 The Equation x2-y2=1 -- 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 x2-my2=N -- Exercises -- Suggested Reading -- Biographies -- 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 -- 13.1 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 -- 14.1 Insolvability of y2=x3+k Using Congruence Considerations -- 14.2 Solving y2=x3+k Using Algebraic Numbers -- 14.3 The Diophantine Equation y(y+1)=x(x+1)(x+2) -- Exercises -- Suggested Reading -- Biographies. |
| Summary |
An introduction to algebraic number theory for senior undergraduates and beginning graduate students in mathematics. It includes numerous examples, and references to further reading and to biographies of mathematicians who have contributed to the development of the subject. Includes over 320 exercises, and an extensive index. |
| Local Note |
eBooks on EBSCOhost EBSCO eBook Subscription Academic Collection - North America |
| Subject |
Algebraic number theory.
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Algebraic number theory. |
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Number theory.
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Number theory. |
| Genre/Form |
Electronic books.
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Einführung.
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Electronic books.
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| Added Author |
Williams, Kenneth S.
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| Other Form: |
Print version: Alaca, Şaban, 1964- Introductory algebraic number theory. Cambridge ; New York : Cambridge University Press, 2004 (DLC) 2003051243 |
| ISBN |
0511166095 (electronic book) |
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9780511166099 (electronic book) |
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9780511164149 |
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0511164149 |
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9780521832502 (hardcover) |
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0521832500 (hardcover) |
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9780521540117 (paperback) |
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0521540119 (paperback) |
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9780511791260 (electronic book) |
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0511791267 (electronic book) |
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9780511164941 (electronic book) |
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0511164947 (electronic book) |
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0511164149 |
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0521832500 (hardcover) |
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0521540119 (paperback) |
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