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. |
Contents
10 Rational Biquadratic Reciprocity | 127 |
CHAPTER 10 | 140 |
Elliptic Curves | 168 |
3 Ramification and Degree | 181 |
Quadratic and Cyclotomic Fields | 188 |
3 The Stickelberger Relation | 213 |
6 Three Applications | 220 |
CHAPTER 15 | 234 |
| 9 | |
| 10 | |
| 11 | |
CHAPTER | 12 |
Applications of Unique Factorization | 17 |
CHAPTER | 18 |
4 The Growth of πx | 23 |
CHAPTER 6 | 26 |
2 Congruence in | 29 |
4 The Chinese Remainder Theorem | 37 |
2 nth Power Residues | 45 |
Quadratic Gauss Sums | 66 |
4 The Sign of the Quadratic Gauss | 73 |
Algebraic Number Theory | 76 |
1 Basic Properties of Finite Fields | 79 |
3 An Application to Quadratic Residues | 85 |
in | 97 |
CHAPTER 9 | 104 |
3 Cubic Residue Character | 112 |
6 The Cubic Character of 2 | 118 |
3 Herbrands Theorem | 241 |
CHAPTER 16 | 247 |
Dirichlet Characters | 253 |
6 Evaluating Ls x at Negative Integers | 261 |
8 | 266 |
x3 + D the Global Case 269 | 269 |
21 | 274 |
39 | 280 |
CHAPTER 19 | 320 |
2 The Group E2E | 323 |
The Weak Dirichlet Unit Theorem | 326 |
4 The Weak MordellWeil Theorem | 328 |
CHAPTER 5 | 333 |
| 337 | |
New Progress in Arithmetic Geometry 1 The Mordell Conjecture 2 Elliptic Curves 3 Modular Curves 4 Heights and the Height Regulator 5 New Resu... | 339 |
Quadratic Reciprocity | 369 |
| 376 | |
| 379 | |
| 386 | |
Other editions - View all
A Classical Introduction to Modern Number Theory Kenneth Ireland,Michael Ira Rosen Limited preview - 1990 |
A Classical Introduction to Modern Number Theory Kenneth Ireland,Michael Ira Rosen Limited preview - 1990 |
Common terms and phrases
a₁ algebraic integers algebraic number field assume b₁ Bernoulli numbers biquadratic Chapter character of order class number coefficients complex numbers congruence conjecture consider Corollary cyclic definition degree denote Dirichlet divides Eisenstein elliptic curve equation Exercise Fermat's finite field Galois Gauss sums implies infinitely many primes irreducible polynomials Jacobi sums law of quadratic Legendre symbol Lemma Let F monic polynomial multiplicative nonresidue nontrivial nonzero number of solutions number theory odd prime P₁ positive integer primary prime ideal prime number PROOF Proposition prove quadratic reciprocity quadratic residue quadratic residue mod rational numbers rational prime reciprocity law relatively prime result follows Riemann hypothesis ring of integers root of unity Section solvable Suppose Theorem unit x₁ y₁ Z/mZ Z/pZ zero zeta function