A Classical Introduction to Modern Number Theory

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Springer Science & Business Media, Sep 7, 1990 - Mathematics - 389 pages
2 Reviews
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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I just read the chapter about the elliptic curve y^2 = x^3 + Dx, it was pretty good. Read full review

Contents

Chapter 1
1
2 Unique Factorization in X
6
3 Unique Factorization in a Principal Ideal Domain
8
4 The Rings Zi and Zw
12
Chapter 2
17
2 Some Arithmetic Functions
18
3 1p Diverges
21
4 The Growth of nx
22
3 Ramification and Degree
181
Chapter 13
188
2 Cyclotomic Fields
193
3 Quadratic Reciprocity Revisited
199
2 The Power Residue Symbol
204
3 The Stickelberger Relation
207
4 The Proof of the Stickelberger Relation
209
5 The Proof of the Eisenstein Reciprocity Law
215

Chapter 3
28
2 Congruence in Z
29
3 The Congruence ax b m
31
4 The Chinese Remainder Theorem
34
Chapter 4
39
2 nth Power Residues
45
Chapter 5
50
2 Law of Quadratic Reciprocity
53
3 A Proof of the Law of Quadratic Reciprocity
58
Chapter 6
66
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
108
1 The Ring Zo
109
2 Residue Class Rings
111
3 Cubic Residue Character
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
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
Chapter 11
151
2 Trace and Norm in Finite Fields
158
3 The Rationality of the Zeta Function Associated to a0x + tfjxT + + attx
161
4 A Proof of the HasseDavenport Relation
163
5 The Last Entry
166
Chapter 12
172
2 Unique Factorization in Algebraic Number Fields
174
6 Three Applications
220
Chapter 15
228
2 Congruences Involving Bernoulli Numbers
234
3 Herbrands Theorem
241
Chapter 16
249
2 A Special Case
251
3 Dirichlet Characters
253
4 Dirichlet Lfunctions
255
5 The Key Step
257
6 Evaluating Ls at Negative Integers
261
Chapter 17
269
2 The Method of Descent
271
3 Legendres Theorem
272
3 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 Equation2 x3 + 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 M y2 x3 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
329
Chapter 20
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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