Quadratic Number Fields

Description

This undergraduate textbook provides an elegant introduction to the arithmetic of quadratic number fields, including many topics not usually covered in books at this level. Quadratic fields offer an introduction to algebraic number theory and some of its central objects: rings of integers, the unit group, ideals and the ideal class group. This textbook provides solid grounding for further study by placing the subject within the greater context of modern algebraic number theory. Going beyond what is usually covered at this level, the book introduces the notion of modularity in the context of quadratic reciprocity, explores the close links between number theory and geometry via Pell conics, and presents applications to Diophantine equations such as the Fermat and Catalan equations as well as elliptic curves. Throughout, the book contains extensive historical comments, numerous exercises (with solutions), and pointers to further study. Assuming a moderate background in elementary number theory and abstract algebra, Quadratic Number Fields offers an engaging first course in algebraic number theory, suitable for upper undergraduate students. Preface Contents 1 Prehistory 1.1 Pythagoras and Euclid 1.2 Diophantus 1.3 Bachet 1.4 Fermat 1.4.1 Integral Solutions of y2 + 2 = x3 1.4.2 The Fermat Equation x4 + y4 = z2 1.5 Euler 1.5.1 The Two-Squares Theorem 1.5.2 Euler's Algebra 1.5.3 Bachet's Equation y2 + 2 = x3 1.5.4 The Cubic Fermat Equation 1.5.5 Euler and the Problem of Units 1.6 Gauss 1.7 Kummer and Dedekind 1.7.1 From Ideal Numbers to Ideals 1.8 Exercises 2 Quadratic Number Fields 2.1 Quadratic Number Fields 2.1.1 Quadratic Extensions as Vector Spaces 2.2 Rings of Integers 2.3 The Unit Circle 2.4 Platon's Hyperbola 2.4.1 Platon's Side and Diagonal Numbers 2.5 Fibonacci's Hyperbola 2.5.1 Generating Functions 2.5.2 Group Law 2.6 Vieta Jumping 2.6.1 The IMO Problem 2.6.2 Markov's Equation 2.6.3 Summary 2.7 Exercises 3 The Modularity Theorem 3.1 Pell Conics Over Fields 3.1.1 Parametrization of Conics 3.1.2 Pell Conics Over Finite Fields 3.2 The Symbols of Legendre, Kronecker, and Jacobi 3.2.1 Kronecker Symbol 3.2.2 Gauss's Lemma 3.2.3 Composite Moduli 3.2.4 Zolotarev and Frobenius 3.2.5 A Few Applications 3.3 Euler's Modularity Conjecture 3.3.1 The Quadratic Reciprocity Law 3.3.2 Proof of Euler's Modularity Conjecture 3.3.3 The Strong Modularity Theorem 3.4 Fp-Rational Points on Curves 3.4.1 Another Proof of the Quadratic Reciprocity Law 3.5 Terjanian's Theorem 3.5.1 Summary 3.6 Exercises 4 Divisibility in Integral Domains 4.1 Units, Primes, and Irreducible Elements 4.1.1 Elements with Prime Norm Are Prime 4.2 Unique Factorization Domains 4.3 Principal Ideal Domains 4.4 Euclidean Domains 4.4.1 Summary 4.5 Exercises 5 Arithmetic in Some Quadratic Number Fields 5.1 The Gaussian Integers 5.1.1 Z[i] Is Norm-Euclidean 5.1.2 Fermat's Last Theorem in Quadratic Number Fields 5.2 The Eisenstein Integers 5.2.1 The Cubic Fermat Equation x3 + y3 + z3 = 0 5.3 The Lucas–Lehmer Test 5.3.1 The Arithmetic in Z[3] 5.3.2 The Lucas–Lehmer Test 5.4 Fermat's Last Theorem for the Exponent 5 5.5 Euclidean Number Fields 5.5.1 Dedekind–Hasse Criterion 5.6 Quadratic Unique Factorization Domains 5.6.1 Euler's Polynomial 5.6.2 Summary 5.7 Exercises 6 Ideals in Quadratic Number Fields 6.1 Motivation 6.1.1 From Ideal Numbers to Ideals 6.1.2 Products of Ideals 6.1.3 The Class Group at Work 6.2 Unique Factorization into Prime Ideals 6.2.1 Classification of Modules 6.2.2 Ideals as Modules 6.2.3 The Cancellation Law 6.2.4 Divisibility of Ideals 6.2.5 Description of Prime Ideals 6.3 Ideal Class Groups 6.3.1 Equivalence of Ideals 6.3.2 Finiteness of the Class Number 6.3.3 Class Group Calculations 6.4 The Diophantine Equation y2 = x3-d 6.4.1 Summary 6.5 Exercises 7 The Pell Equation 7.1 The Solvability of the Pell Equation 7.1.1 The Negative Pell Equation 7.2 Which Numbers Are Norms? 7.2.1 Davenport's Lemma 7.3 Computing the Solution of the Pell Equation 7.4 Parametrized Units 7.5 Factorization Algorithms 7.6 Diophantine Equations 7.6.1 Summary 7.7 Exercises 8 Catalan's Equation 8.1 Lebesgue's Theorem 8.2 Euler's Theorem 8.2.1 Monsky's Proof 8.3 The Theorems of Størmer and Ko Chao 8.3.1 Application to Catalan's Equation 8.4 Euler's Equation via Pure Cubic Number Fields 8.4.1 Units in Pure Cubic Number Fields 8.4.2 The Equation x3 + 2y3 = 1 8.4.3 The Theorem of Delaunay and Nagell 8.5 Mihailescu's Proof 8.5.1 Summary 8.6 Exercises 9 Ambiguous Ideal Classes and Quadratic Reciprocity 9.1 Ambiguous Ideal Classes 9.1.1 Exact Sequences 9.1.2 Ambiguous Ideal Classes 9.1.3 Hilbert's Theorem 90 9.2 The Ambiguous Class Number Formula 9.3 The Quadratic Reciprocity Law 9.3.1 Summary 9.4 Exercises 10 Quadratic Gauss Sums 10.1 Dirichlet Characters 10.1.1 Primitive Characters 10.1.2 The Character Group of Finite abelian Groups 10.1.3 Classification of Quadratic Dirichlet Characters 10.1.4 Modularity and Reciprocity 10.2 Pell Forms 10.3 Fekete Polynomials 10.3.1 Gauss's Sixth Proof 10.4 The Analytic Class Number Formula 10.5 Modularity 10.5.1 Modularity of Polynomials 10.5.2 Modularity of Number Fields 10.5.3 Pell Forms 10.6 Modularity of Elliptic Curves 10.6.1 Group Law 10.6.2 Curves with Complex Multiplication 10.6.3 Hasse's Theorem 10.6.4 Modularity of Elliptic Curves 10.7 Exercises A Computing with Pari and Sage A.1 Pari A.1.1 Arithmetic in Integers A.1.2 Arithmetic in Quadratic Number Fields A.2 Sage A.2.1 Number Fields A.2.2 Elliptic Curves B Solutions Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Bibliography Name Index Subject Index