A Classical Introduction to Modern Number Theory

Couverture
Springer Science & Business Media, 17 avr. 2013 - 394 pages
Bridging the gap between elementary number theory and the systematic study of advanced topics, A Classical Introduction to Modern Number Theory is a well-developed and accessible text that requires only a familiarity with basic abstract algebra. Historical development is stressed throughout, along with wide-ranging coverage of significant results with comparatively elementary proofs, some of them new. An extensive bibliography and many challenging exercises are also included. This second edition has been corrected and contains two new chapters which provide a complete proof of the Mordell-Weil theorem for elliptic curves over the rational numbers, and an overview of recent progress on the arithmetic of elliptic curves.
 

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Table des matières

3 Unique Factorization in a Principal Ideal Domain
8
CHAPTER
17
4 The Growth of Trx
23
CHAPTER 6
26
CHAPTER 11
36
CHAPTER 4
39
2 Law of Quadratic Reciprocity
53
Quadratic Gauss Sums
66
3 The Stickelberger Relation
207
5 The Proof of the Eisenstein Reciprocity Law
215
CHAPTER 15
223
2 Congruences Involving Bernoulli Numbers
234
3 Herbrands Theorem
241
CHAPTER 16
247
3 Dirichlet Characters
253
6 Evaluating Ls X at Negative Integers
261

2 The Quadratic Character of 2
69
3 Quadratic Gauss Sums
70
4 The Sign of the Quadratic Gauss Sum
73
CHAPTER 7
79
2 The Existence of Finite Fields
83
3 An Application to Quadratic Residues
85
CHAPTER 8
88
2 Gauss Sums
91
3 Jacobi Sums
92
4 The Equation x + y 1 in F
97
5 More on Jacobi Sums
98
6 Applications
101
7 A General Theorem
102
CHAPTER 9
104
Cubic and Biquadratic Reciprocity
108
1 The Ring Zo
109
2 Residue Class Rings
111
3 Cubic Residue Character 12
112
4 Proof of the Law of Cubic Reciprocity
115
5 Another Proof of the Law of Cubic Reciprocity
117
6 The Cubic Character of 2 18
118
Preliminaries
119
8 The Quartic Residue Symbol
121
9 The Law of Biquadratic Reciprocity
123
10 Rational Biquadratic Reciprocity
127
11 The Constructibility of Regular Polygons
130
12 Cubic Gauss Sums and the Problem of Kummer
131
CHAPTER 10
138
2 Chevalleys Theorem
143
3 Gauss and Jacobi Sums over Finite Fields
145
The Zeta Function
151
2 Trace and Norm in Finite Fields
158
5 The Last Entry
166
CHAPTER 12
172
3 Ramification and Degree
181
Quadratic and Cyclotomic Fields
188
3 Quadratic Reciprocity Revisited
199
Diophantine Equations
269
2 The Method of Descent
271
3 Legendres Theorem
272
4 Sophie Germains Theorem
275
5 Pells Equation
276
6 Sums of Two Squares
278
7 Sums of Four Squares
280
Exponent 3
284
9 Cubic Curves with Infinitely Many Rational Points
287
10 The Equation y2 x + k
288
11 The First Case of Fermats Conjecture for Regular Exponent
290
12 Diophantine Equations and Diophantine Approximation
292
CHAPTER 18
297
2 Local and Global Zeta Functions of an Elliptic Curve
301
3 y2 x3 + D the Local Case
304
4 y x Dx the Local Case
306
5 Hecke Lfunctions
307
6 y x Dx the Global Case
310
7 y2 x3 + D the Global Case
312
8 Final Remarks
314
CHAPTER 19
319
2 The Group E2E
323
3 The Weak Dirichlet Unit Theorem
326
4 The Weak MordellWeil Theorem
328
5 The Descent Argument
330
New Progress in Arithmetic Geometry
339
1 The Mordell Conjecture
340
2 Elliptic Curves
343
3 Modular Curves
345
4 Heights and the Height Regulator
347
5 New Results on the BirchSwinnertonDyer Conjecture
353
6 Applications to Gausss Class Number Conjecture
358
Selected Hints for the Exercises
367
Bibliography
375
Index
385
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