9780387973296

A Classical Introduction to Modern Number Theory

by ;
  • ISBN13:

    9780387973296

  • ISBN10:

    038797329X

  • Edition: 2nd
  • Format: Hardcover
  • Copyright: 1991-01-01
  • Publisher: Springer Verlag

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Summary

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.

Table of Contents

Preface to the Second Edition v
Preface vii
Unique Factorization
1(16)
Unique Factorization in Z
1(5)
Unique Factorization in k[x]
6(2)
Unique Factorization in a Principal Ideal Domain
8(4)
The Rings Z[i] and Z[ω]
12(5)
Applications of Unique Factorization
17(11)
Infinitely Many Primes in Z
17(1)
Some Arithmetic Functions
18(3)
&Sigma 1/p Diverges
21(1)
The Growth of &pi(x)
22(6)
Congruence
28(11)
Elementary Observations
28(1)
Congruence in Z
29(2)
The Congruence ax ≡ b(m)
31(3)
The Chinese Remainder Theorem
34(5)
The Structure of U (Z/nZ)
39(11)
Primitive Roots and the Group Structure of U (Z/nZ)
39(6)
nth Power Residues
45(5)
Quadratic Reciprocity
50(16)
Quadratic Residues
50(3)
Law of Quadratic Reciprocity
53(5)
A Proof of the Law of Quadratic Reciprocity
58(8)
Quadratic Gauss Sums
66(13)
Algebraic Numbers and Algebraic Integers
66(3)
The Quadratic Character of 2
69(1)
Quadratic Gauss Sums
70(3)
The Sign of the Quadratic Gauss Sum
73(6)
Finite Fields
79(9)
Basic Properties of Finite Fields
79(4)
The Existence of Finite Fields
83(2)
An Application to Quadratic Residues
85(3)
Gauss and Jacobi Sums
88(20)
Multiplicative Characters
88(3)
Gauss Sums
91(1)
Jacobi Sums
92(5)
The Equation xn + yn = 1 in Fp
97(1)
More on Jacobi Sums
98(3)
Applications
101(1)
A General Theorem
102(6)
Cubic and Biquadratic Reciprocity
108(30)
The Ring Z[ω]
109(2)
Residue Class Rings
111(1)
Cubic Residue Character
112(3)
Proof of the Law of Cubic Reciprocity
115(2)
Another Proof of the Law of Cubic Reciprocity
117(1)
The Cubic Character of 2
118(1)
Biquadratic Reciprocity: Preliminaries
119(2)
The Quartic Residue Symbol
121(2)
The Law of Biquadratic Reciprocity
123(4)
Rational Biquadratic Reciprocity
127(3)
The Constructibility of Regular Polygons
130(1)
Cubic Gauss Sums and the Problem of Kummer
131(7)
Equations over Finite Fields
138(13)
Affine Space, Projective Space, and Polynomials
138(5)
Chevalley's Theorem
143(2)
Gauss and Jacobi Sums over Finite Fields
145(6)
The Zeta Function
151(21)
The Zeta Function of a Projective Hypersurface
151(7)
Trace and Norm in Finite Fields
158(3)
The Rationality of the Zeta Function Associated to a0x0m + a1x1m +...+ anxnm
161(2)
A Proof of the Hasse-Davenport Relation
163(3)
The Last Entry
166(6)
Algebraic Number Theory
172(16)
Algebraic Preliminaries
172(2)
Unique Factorization in Algebraic Number Fields
174(7)
Ramification and Degree
181(7)
Quadratic and Cyclotomic Fields
188(15)
Quadratic Number Fields
188(5)
Cyclotomic Fields
193(6)
Quadratic Reciprocity Revisited
199(4)
The Stickelberger Relation and the Eisenstein Reciprocity Law
203(25)
The Norm of an Ideal
203(1)
The Power Residue Symbol
204(3)
The Stickelberger Relation
207(2)
The Proof of the Stickelberger Relation
209(6)
The Proof of the Eisenstein Reciprocity Law
215(5)
Three Applications
220(8)
Bernoulli Numbers
228(21)
Bernoulli Numbers: Definitions and Applications
228(6)
Congruences Involving Bernoulli Numbers
234(7)
Herbrand's Theorem
241(8)
Dirichlet L-functions
249(20)
The Zeta Function
249(2)
A Special Case
251(2)
Dirichlet Characters
253(2)
Dirichlet L-functions
255(2)
The Key Step
257(4)
Evaluating L(s, x) at Negative Integers
261(8)
Diophantine Equations
269(28)
Generalities and First Examples
269(2)
The Method of Descent
271(1)
Legendre's Theorem
272(3)
Sophie Germain's Theorem
275(1)
Pell's Equation
276(2)
Sums of Two Squares
278(2)
Sums of Four Squares
280(4)
The Fermat Equation: Exponent 3
284(3)
Cubic Curves with Infinitely Many Rational Points
287(1)
The Equation y2 = x3 + k
288(2)
The First Case of Fermat's Conjecture for Regular Exponent
290(2)
Diophantine Equations and Diophantine Approximation
292(5)
Elliptic Curves
297(22)
Generalities
297(4)
Local and Global Zeta Functions of an Elliptic Curve
301(3)
y2 = x3 + D, the Local Case
304(2)
y2 = x3 - Dx, the Local Case
306(1)
Hecke L-functions
307(3)
y2 = x3 - Dx, the Global Case
310(2)
y2 = x3 + D, the Global Case
312(2)
Final Remarks
314(5)
The Mordell - Weil Theorem
319(20)
The Addition Law and Several Identities
320(3)
The Group E/2E
323(3)
The Weak Dirichlet Unit Theorem
326(2)
The Weak Mordell-Weil Theorem
328(2)
The Descent Argument
330(9)
New Progress in Arithmetic Geometry
339(28)
The Mordell Conjecture
340(3)
Elliptic Curves
343(2)
Modular Curves
345(3)
Heights and the Height Regulator
348(5)
New Results on the Birch-Swinnerton-Dyer Conjecture
353(5)
Applications to Gauss's Class Number Conjecture
358(9)
Selected Hints for the Exercises 367(8)
Bibliography 375(10)
Index 385

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