## A Classical Introduction to Modern Number TheoryBridging 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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#### Review: A Classical Introduction to Modern Number Theory (Graduate Texts in Mathematics) (Graduate Texts in Mathematics #84)

User Review - Dan - GoodreadsI 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 |

385 | |

### Common terms and phrases

algebraic integers algebraic number field analytic arithmetic assertion assume automorphism Bernoulli numbers biquadratic Chapter character of order class number coefficients complex numbers congruence conjecture consider cyclic definition degree denote Dirichlet character divides Eisenstein Exercise exist fact finite field formula Gauss sums Geometry give Hecke character hypersurface implies infinitely many primes integral solution irreducible polynomials isomorphic Jacobi sums L-functions law of quadratic Legendre symbol Lemma modular Mordell multiplicative nontrivial nonzero number of elements number of points number of solutions number theory odd prime ordp points at infinity positive integer primary prime ideal prime number primitive root Proposition prove q elements quadratic number fields quadratic reciprocity quadratic residue rational numbers rational points rational prime reciprocity law relatively prime result follows Riemann hypothesis ring of integers root of unity Section solvable square square-free subgroup Suppose unique factorization unit write Z/pZ zero zeta function