A Classical Introduction to Modern Number Theory

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Springer Science & Business Media, 9 mars 2013 - 344 pages
This book is a revised and greatly expanded version of our book Elements of Number Theory published in 1972. As with the first book the primary audience we envisage consists of upper level undergraduate mathematics majors and graduate students. We have assumed some familiarity with the material in a standard undergraduate course in abstract algebra. A large portion of Chapters 1-11 can be read even without such background with the aid of a small amount of supplementary reading. The later chapters assume some knowledge of Galois theory, and in Chapters 16 and 18 an acquaintance with the theory of complex variables is necessary. Number theory is an ancient subject and its content is vast. Any intro ductory book must, of necessity, make a very limited selection from the fascinat ing array of possible topics. Our focus is on topics which point in the direction of algebraic number theory and arithmetic algebraic geometry. By a careful selection of subject matter we have found it possible to exposit some rather advanced material without requiring very much in the way oftechnical background. Most of this material is classical in the sense that is was dis covered during the nineteenth century and earlier, but it is also modern because it is intimately related to important research going on at the present time.
 

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Table des matières

CHAPTER
1
Unique Factorization in a Principal Ideal Domain
8
CHAPTER
14
Applications of Unique Factorization
17
CHAPTER
18
The Growth of tx
23
Congruence in Z
29
CHAPTER 4
35
CHAPTER 15
223
Bernoulli Numbers
228
Congruences Involving Bernoulli Numbers
234
Herbrands Theorem
241
CHAPTER 16
248
Dirichlet Lfunctions
249
A Special Case
251
Dirichlet Characters
253

CHAPTER 5
50
CHAPTER 6
66
CHAPTER 7
79
Jacobi Sums
92
Cubic and Biquadratic Reciprocity
108
Proof of the Law of Cubic Reciprocity
115
The Quartic Residue Symbol
121
Rational Biquadratic Reciprocity
127
CHAPTER 10
135
CHAPTER 11
147
Trace and Norm in Finite Fields
158
The Last Entry
166
Algebraic Number Theory
172
CHAPTER 13
186
The Stickelberger Relation and the Eisenstein Reciprocity Law
203
The Power Residue Symbol
204
The Stickelberger Relation
207
The Proof of the Stickelberger Relation
209
The Proof of the Eisenstein Reciprocity Law
215
Three Applications
220
Dirichlet Lfunctions
255
The Key Step
257
Evaluating Ls X at Negative Integers
261
CHAPTER 17
269
The Method of Descent
271
Legendres Theorem
272
Sophie Germains Theorem
275
Pells Equation
276
Sums of Two Squares
278
Sums of Four Squares
280
Exponent 3
284
Cubic Curves with Infinitely Many Rational Points
287
The Equation y x + k
288
The First Case of Fermats Conjecture for Regular Exponent
290
Diophantine Equations and Diophantine Approximation
292
Elliptic Curves
297
Selected Hints for the Exercises
319
Index
337
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