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 welldeveloped and accessible text that requires only a familiarity with basic abstract algebra. Historical development is stressed throughout, along with wideranging 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 MordellWeil 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
Review: A Classical Introduction to Modern Number Theory (Graduate Texts in Mathematics) (Graduate Texts in Mathematics #84)
User Review  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 Lfunctions 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 squarefree subgroup Suppose unique factorization unit write Z/pZ zero zeta function