Introductory Algebraic Number Theory,9780521540117

Introductory Algebraic Number Theory

by
ISBN13: 9780521540117
ISBN10: 0521540119
Format: Paperback
Pub. Date: 2003-11-17
Publisher(s): Cambridge University Press
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Summary

Suitable for senior undergraduates and beginning graduate students in mathematics, this book is an introduction to algebraic number theory at an elementary level. Prerequisites are kept to a minimum, and numerous examples illustrating the material occur throughout the text. References to suggested readings and to the biographies of mathematicians who have contributed to the development of algebraic number theory are provided at the end of each chapter. Other features include over 320 exercises, an extensive index, and helpful location guides to theorems in the text.

Table of Contents

List of Tables xi
Notation xiii
Introduction xv
1 Integral Domains 1(26)
1.1 Integral Domains
1(4)
1.2 Irreducibles and Primes
5(3)
1.3 Ideals
8(2)
1.4 Principal Ideal Domains
10(6)
1.5 Maximal Ideals and Prime Ideals
16(5)
1.6 Sums and Products of Ideals
21(2)
Exercises
23(2)
Suggested Reading
25(1)
Biographies
25(2)
2 Euclidean Domains 27(27)
2.1 Euclidean Domains
27(3)
2.2 Examples of Euclidean Domains
30(7)
2.3 Examples of Domains That are Not Euclidean
37(9)
2.4 Almost Euclidean Domains
46(1)
2.5 Representing Primes by Binary Quadratic Forms
47(2)
Exercises
49(2)
Suggested Reading
51(2)
Biographies
53(1)
3 Noetherian Domains 54(20)
3.1 Noetherian Domains
54(3)
3.2 Factorization Domains
57(3)
3.3 Unique Factorization Domains
60(4)
3.4 Modules
64(3)
3.5 Noetherian Modules
67(4)
Exercises
71(1)
Suggested Reading
72(1)
Biographies
73(1)
4 Elements Integral over a Domain 74(14)
4.1 Elements Integral over a Domain
74(7)
4.2 Integral Closure
81(5)
Exercises
86(1)
Suggested Reading
87(1)
Biographies
87(1)
5 Algebraic Extensions of a Field 88(21)
5.1 Minimal Polynomial of an Element Algebraic over a Field
88(2)
5.2 Conjugates of α over Κ
90(1)
5.3 Conjugates of an Algebraic Integer
91(3)
5.4 Algebraic Integers in a Quadratic Field
94(4)
5.5 Simple Extensions
98(4)
5.6 Multiple Extensions
102(4)
Exercises
106(2)
Suggested Reading
108(1)
Biographies
108(1)
6 Algebraic Number Fields 109(32)
6.1 Algebraic Number Fields
109(3)
6.2 Conjugate Fields of an Algebraic Number Field
112(4)
6.3 The Field Polynomial of an Element of an Algebraic Number Field
116(7)
6.4 The Discriminant of a set of Elements in an Algebraic Number Field
123(6)
6.5 Basis of an Ideal
129(8)
6.6 Prime Ideals in Rings of Integers
137(1)
Exercises
138(2)
Suggested Reading
140(1)
Biographies
140(1)
7 Integral Bases 141(53)
7.1 Integral Basis of an Algebraic Number Field
141(19)
7.2 Minimal Integers
160(10)
7.3 Some Integral Bases in Cubic Fields
170(8)
7.4 Index and Minimal Index of an Algebraic Number Field
178(8)
7.5 Integral Basis of a Cyclotomic Field
186(3)
Exercises
189(2)
Suggested Reading
191(2)
Biographies
193(1)
8 Dedekind Domains 194(24)
8.1 Dedekind Domains
194(1)
8.2 Ideals in a Dedekind Domain
195(5)
8.3 Factorization into Prime Ideals
200(6)
8.4 Order of an Ideal with Respect to a Prime Ideal
206(9)
8.5 Generators of Ideals in a Dedekind Domain
215(1)
Exercises
216(1)
Suggested Reading
217(1)
9 Norms of Ideals 218(18)
9.1 Norm of an Integral Ideal
218(4)
9.2 Norm and Trace of an Element
222(6)
9.3 Norm of a Product of Ideals
228(3)
9.4 Norm of a Fractional Ideal
231(2)
Exercises
233(1)
Suggested Reading
234(1)
Biographies
235(1)
10 Factoring Primes in a Number Field 236(28)
10.1 Norm of a Prime Ideal
236(5)
10.2 Factoring Primes in a Quadratic Field
241(8)
10.3 Factoring Primes in a Monogenic Number Field
249(4)
10.4 Some Factorizations in Cubic Fields
253(4)
10.5 Factoring Primes in an Arbitrary Number Field
257(3)
10.6 Factoring Primes in a Cyclotomic Field
260(1)
Exercises
261(1)
Suggested Reading
262(2)
11 Units in Real Quadratic Fields 264(35)
11.1 The Units of Z+Z/2
264(3)
11.2 The Equation x2-y2=1
267(4)
11.3 Units of Norm 1
271(4)
11.4 Units of Norm -1
275(3)
11.5 The Fundamental Unit
278(8)
11.6 Calculating the Fundamental Unit
286(8)
11.7 The Equation x2-my2=N
294(3)
Exercises
297(1)
Suggested Reading
298(1)
Biographies
298(1)
12 The Ideal Class Group 299(45)
12.1 Ideal Class Group
299(1)
12.2 Minkowski's Translate Theorem
300(5)
12.3 Minkowski's Convex Body Theorem
305(1)
12.4 Minkowski's Linear Forms Theorem
306(5)
12.5 Finiteness of the Ideal Class Group
311(3)
12.6 Algorithm to Determine the Ideal Class Group
314(17)
12.7 Applications to Binary Quadratic Forms
331(10)
Exercises
341(2)
Suggested Reading
343(1)
Biographies
343(1)
13 Dirichlet's Unit Theorem 344(41)
13.1 Valuations of an Element of a Number Field
344(2)
13.2 Properties of Valuations
346(13)
13.3 Proof of Dirichlet's Unit Theorem
359(2)
13.4 Fundamental System of Units
361(2)
13.5 Roots of Unity
363(6)
13.6 Fundamental Units in Cubic Fields
369(9)
13.7 Regulator
378(4)
Exercises
382(1)
Suggested Reading
383(1)
Biographies
384(1)
14 Applications to Diophantine Equations 385(28)
14.1 Insolvability of y2=x3+k Using Congruence Considerations
385(4)
14.2 Solving y2=x3+k Using Algebraic Numbers
389(12)
14.3 The Diophantine Equation y(y+1)=x(x+1)(x+2)
401(9)
Exercises
410(1)
Suggested Reading
411(1)
Biographies
411(2)
List of Definitions 413(4)
Location of Theorems 417(4)
Location of Lemmas 421(2)
Bibliography 423(2)
Index 425

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