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Problems in Algebraic Number Theory

M. Ram Murty and Jody Esmonde
Publisher: 
Springer-Verlag
Publication Date: 
2005
Number of Pages: 
346
Format: 
Hardcover
Edition: 
2
Series: 
Graduate Texts in Mathematics
Price: 
59.95
ISBN: 
0387221824
Category: 
Textbook
[Reviewed by
Álvaro Lozano-Robledo
, on
02/5/2005
]

Most professors would agree that a student who truly wants to learn any given area of mathematics will benefit enormously from solving many problems on the subject at hand. Struggling with a problem often helps one understand the finest aspects of the theory. However, can a student learn an area of mathematics just by solving a systematic collection of problems?

Problems in Algebraic Number Theory is intended to be used by the student for independent study of the subject. It provides the reader with a large collection of problems (about 500), at the level of a first course on the algebraic theory of numbers (with undergraduate algebra as a prerequisite). The volume also includes completely spelled-out solutions to all exercises. The list of topics include elementary number theory, algebraic numbers and number fields, Dedekind domains, ideal class groups, structure of the unit group, reciprocity laws (quadratic and higher) and Dirichlet L-functions. A new chapter on density theorems was added in the second edition. Each section starts with some basic definitions and theorems (with proofs), and a couple of solved examples.

The reviewer thinks that the authors have done a fantastic job choosing the problems, which are perfectly arranged so the students can progressively move on from topic to topic, discovering on their own the proofs of the most well-known results and applications. The exposition of the solutions is very clear and helps to introduce different important techniques.

However, the reviewer believes that a student is better served with a healthy balance between traditional lectures and problem-solving. Without proper advising the student might miss the 'soul' of the subject. The book feels at times like a bare sequence of definitions, theorems and exercises. The readers are supposed to find out, on their own through problem solving, why the concepts are introduced, and this might be a path that not everybody can easily follow. The student might find hard to differentiate routine problems from the more relevant exercises which will be used as lemmas in consequent sections. It is the opinion of the reviewer that each chapter should be complemented with a couple of lectures by a professor to emphasize the main ideas and clarify the goals of the developing theory. In any case, the book is an excellent resource for the instructor and the student as a companion to any algebraic number theory course.


Álvaro Lozano-Robledo is H. C. Wang Assistant Professor at Cornell University.

 

Preface to the Second Edition vii
Preface to the First Edition ix
Acknowledgments xi
I Problems
1 Elementary Number Theory 3
1.1 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Applications of Unique Factorization . . . . . . . . . . . . . 8
1.3 The ABC Conjecture . . . . . . . . . . . . . . . . . . . . . 9
1.4 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 10
2 Euclidean Rings 13
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Gaussian Integers . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Eisenstein Integers . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Some Further Examples . . . . . . . . . . . . . . . . . . . . 21
2.5 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 25
3 Algebraic Numbers and Integers 27
3.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Liouville's Theorem and Generalizations . . . . . . . . . . . 29
3.3 Algebraic Number Fields . . . . . . . . . . . . . . . . . . . . 32
3.4 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 38
4 Integral Bases 41
4.1 The Normand the Trace . . . . . . . . . . . . . . . . . . . . 41
4.2 Existence of an Integral Basis . . . . . . . . . . . . . . . . . 43
4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.4 Ideals in OK . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.5 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 50
xiii
xiv CONTENTS
5 Dedekind Domains 53
5.1 Integral Closure . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2 Characterizing Dedekind Domains . . . . . . . . . . . . . . 55
5.3 Fractional Ideals and Unique Factorization . . . . . . . . . . 57
5.4 Dedekind's Theorem . . . . . . . . . . . . . . . . . . . . . . 63
5.5 Factorization in OK . . . . . . . . . . . . . . . . . . . . . . 65
5.6 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 66
6 The Ideal Class Group 69
6.1 Elementary Results . . . . . . . . . . . . . . . . . . . . . . . 69
6.2 Finiteness of the Ideal Class Group . . . . . . . . . . . . . . 71
6.3 Diophantine Equations . . . . . . . . . . . . . . . . . . . . . 73
6.4 Exponents of Ideal Class Groups . . . . . . . . . . . . . . . 75
6.5 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 76
7 Quadratic Reciprocity 81
7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 81
7.2 Gauss Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.3 The Law of Quadratic Reciprocity . . . . . . . . . . . . . . 86
7.4 Quadratic Fields . . . . . . . . . . . . . . . . . . . . . . . . 88
7.5 Primes in Special Progressions . . . . . . . . . . . . . . . . 91
7.6 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 94
8 The Structure of Units 99
8.1 Dirichlet's Unit Theorem . . . . . . . . . . . . . . . . . . . 99
8.2 Units in Real Quadratic Fields . . . . . . . . . . . . . . . . 108
8.3 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 115
9 Higher Reciprocity Laws 117
9.1 Cubic Reciprocity . . . . . . . . . . . . . . . . . . . . . . . 117
9.2 Eisenstein Reciprocity . . . . . . . . . . . . . . . . . . . . . 122
9.3 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 125
10 Analytic Methods 127
10.1 The Riemann and Dedekind Zeta Functions . . . . . . . . . 127
10.2 Zeta Functions of Quadratic Fields . . . . . . . . . . . . . . 130
10.3 Dirichlet's L-Functions . . . . . . . . . . . . . . . . . . . . . 133
10.4 Primes in Arithmetic Progressions . . . . . . . . . . . . . . 134
10.5 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 136
11 Density Theorems 139
11.1 Counting Ideals in a Fixed Ideal Class . . . . . . . . . . . . 139
11.2 Distribution of Prime Ideals . . . . . . . . . . . . . . . . . . 146
11.3 The Chebotarev density theorem . . . . . . . . . . . . . . . 150
11.4 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 153
CONTENTS xv
II Solutions
1 Elementary Number Theory 159
1.1 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
1.2 Applications of Unique Factorization . . . . . . . . . . . . . 166
1.3 The ABC Conjecture . . . . . . . . . . . . . . . . . . . . . 170
1.4 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 173
2 Euclidean Rings 179
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 179
2.2 Gaussian Integers . . . . . . . . . . . . . . . . . . . . . . . . 181
2.3 Eisenstein Integers . . . . . . . . . . . . . . . . . . . . . . . 185
2.4 Some Further Examples . . . . . . . . . . . . . . . . . . . . 187
2.5 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 189
3 Algebraic Numbers and Integers 197
3.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 197
3.2 Liouville's Theorem and Generalizations . . . . . . . . . . . 198
3.3 Algebraic Number Fields . . . . . . . . . . . . . . . . . . . . 199
3.4 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 202
4 Integral Bases 207
4.1 The Normand the Trace . . . . . . . . . . . . . . . . . . . . 207
4.2 Existence of an Integral Basis . . . . . . . . . . . . . . . . . 208
4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
4.4 Ideals in OK . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
4.5 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 214
5 Dedekind Domains 227
5.1 Integral Closure . . . . . . . . . . . . . . . . . . . . . . . . . 227
5.2 Characterizing Dedekind Domains . . . . . . . . . . . . . . 228
5.3 Fractional Ideals and Unique Factorization . . . . . . . . . . 229
5.4 Dedekind's Theorem . . . . . . . . . . . . . . . . . . . . . . 233
5.5 Factorization in OK . . . . . . . . . . . . . . . . . . . . . . 234
5.6 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 235
6 The Ideal Class Group 245
6.1 Elementary Results . . . . . . . . . . . . . . . . . . . . . . . 245
6.2 Finiteness of the Ideal Class Group . . . . . . . . . . . . . . 245
6.3 Diophantine Equations . . . . . . . . . . . . . . . . . . . . . 247
6.4 Exponents of Ideal Class Groups . . . . . . . . . . . . . . . 248
6.5 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 250
xvi CONTENTS
7 Quadratic Reciprocity 263
7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 263
7.2 Gauss Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
7.3 The Law of Quadratic Reciprocity . . . . . . . . . . . . . . 267
7.4 Quadratic Fields . . . . . . . . . . . . . . . . . . . . . . . . 270
7.5 Primes in Special Progressions . . . . . . . . . . . . . . . . 270
7.6 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 272
8 The Structure of Units 279
8.1 Dirichlet's Unit Theorem . . . . . . . . . . . . . . . . . . . 279
8.2 Units in Real Quadratic Fields . . . . . . . . . . . . . . . . 284
8.3 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 291
9 Higher Reciprocity Laws 299
9.1 Cubic Reciprocity . . . . . . . . . . . . . . . . . . . . . . . 299
9.2 Eisenstein Reciprocity . . . . . . . . . . . . . . . . . . . . . 303
9.3 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 308
10 Analytic Methods 313
10.1 The Riemann and Dedekind Zeta Functions . . . . . . . . . 313
10.2 Zeta Functions of Quadratic Fields . . . . . . . . . . . . . . 316
10.3 Dirichlet's L-Functions . . . . . . . . . . . . . . . . . . . . . 320
10.4 Primes in Arithmetic Progressions . . . . . . . . . . . . . . 322
10.5 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 324
11 Density Theorems 333
11.1 Counting Ideals in a Fixed Ideal Class . . . . . . . . . . . . 333
11.2 Distribution of Prime Ideals . . . . . . . . . . . . . . . . . . 337
11.3 The Chebotarev density theorem . . . . . . . . . . . . . . . 340
11.4 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 341
Bibliography 347
Index 349

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