# A Classical Introduction to Modern Number Theory

Springer Science & Business Media, Sep 7, 1990 - Mathematics - 389 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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### 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 Copyright

### About the author (1990)

Michael Rosen started writing as a teenager, when his mother needed some poems for Radio programs she was making. While at college, he wrote a play which was staged at the Royal Court theatre in London. Rosen's first book was published in 1974, and he is one of Britain's leading children's poets. Michael Rosen launched the National Year of Literacy project, which encouraged children to help produce an Anthology to be used during the Literacy Hour in primary schools. Children ages 4-11 were invited to submit poems and illustrations featuring their favorite tree. Rosen also led the final judging sessions to decide which submissions would be included.