Quadratic Number Theory: An Invitation to Algebraic Methods in the Higher Arithmetic (Dolciani Mathematical Expositions) (Dolciani Mathematical Expositions, 52)

Quadratic Number Theory is an introduction to algebraic number theory for readers with a moderate knowledge of elementary number theory and some familiarity with the terminology of abstract algebra. By restricting attention to questions about squares the author achieves the dual goals of making the presentation accessible to undergraduates and reflecting the historical roots of the subject. The representation of integers by quadratic forms is emphasized throughout the text. Lehman introduces an innovative notation for ideals of a quadratic domain that greatly facilitates computation and he uses this to particular effect. The text has an unusual focus on actual computation. This focus, and this notation, serve the author's historical purpose as well; ideals can be seen as number-like objects, as Kummer and Dedekind conceived of them. The notation can be adapted to quadratic forms and provides insight into the connection between quadratic forms and ideals. The computation of class groups and continued fraction representations are featured—the author's notation makes these computations particularly illuminating. Quadratic Number Theory, with its exceptionally clear prose, hundreds of exercises, and historical motivation, would make an excellent textbook for a second undergraduate course in number theory. The clarity of the exposition would also make it a terrific choice for independent reading. It will be exceptionally useful as a fruitful launching pad for undergraduate research projects in algebraic number theory. 1991 Mathematics Subject Classification. Cover 1 Title page 2 Copyright 3 Contents 4 Preface 8 Acknowledgments 14 Introduction: A Brief Review of Elementary Number Theory 16 0.1. Linear Equations and Congruences 16 0.2. Quadratic Congruences Modulo Primes 21 0.3. Quadratic Congruences Modulo Composite Integers 25 Part One: Quadratic Domains and Ideals 30 Chapter 1. Gaussian Integers and Sums of Two Squares 32 1.1. Sums of Two Squares 32 1.2. Gaussian Integers 40 1.3. Ideal Form for Gaussian Integers 47 1.4. Factorization and Multiplication with Ideal Forms 53 1.5. Reduction of Ideal Forms for Gaussian Integers 59 1.6. Sums of Two Squares Revisited 63 Gaussian Integers and Sums of Two Squares—Review 67 Chapter 2. Quadratic Domains 70 2.1. Quadratic Numbers and Quadratic Integers 71 2.2. Domains of Quadratic Integers 76 2.3. Ideal Form for Quadratic Integers 83 2.4. Ideal Numbers 89 2.5. Quadratic Domains with Unique Factorization 95 2.6. Quadratic Domains without Unique Factorization 101 Quadratic Domains—Review 106 Chapter 3. Ideals of Quadratic Domains 108 3.1. Ideals and Ideal Numbers 109 3.2. Writing Ideals as Ideal Numbers 114 3.3. Prime Ideals of Quadratic Domains 118 3.4. Multiplication of Ideals 122 3.5. Prime Ideal Factorization 127 3.6. A Formula for Ideal Multiplication 134 Ideals of Quadratic Domains—Review 141 Part Two: Quadratic Forms and Ideals 144 Chapter 4. Quadratic Forms 146 4.1. Classification of Quadratic Forms 146 4.2. Equivalence of Quadratic Forms 151 4.3. Representations of Integers by Quadratic Forms 155 4.4. Genera of Quadratic Forms 160 Quadratic Forms—Review 166 Chapter 5. Correspondence between Forms and Ideals 168 5.1. Equivalence of Ideals 169 5.2. Quadratic Forms Associated to an Ideal 173 5.3. Composition of Binary Quadratic Forms 179 5.4. Class Groups of Ideals and Quadratic Forms 185 Correspondence between Forms and Ideals—Review 190 Part Three: Positive Definite Quadratic Forms 192 Chapter 6. Class Groups of Negative Discriminant 194 6.1. Reduced Positive Definite Quadratic Forms 194 6.2. Calculation of Ideal Class Groups 201 6.3. Genera of Ideal Classes 205 Class Groups of Negative Discriminant—Review 212 Chapter 7. Representations by Positive Definite Forms 214 7.1. Negative Discriminants with Trivial Class Groups 215 7.2. Principal Square Domains 220 7.3. Quadratic Domains that Are Not Principal Square Domains 227 7.4. Construction of Representations 234 Representations by Positive Definite Forms—Review 240 Chapter 8. Class Groups of Quadratic Subdomains 242 8.1. Constructing Class Groups of Subdomains 242 8.2. Projection Homomorphisms 249 8.3. The Kernel of a Projection Homomorphism 254 Class Groups of Quadratic Subdomains—Review 259 Part Four: Indefinite Quadratic Forms 260 Chapter 9. Continued Fractions 262 9.1. Introduction to Continued Fractions 262 9.2. Pell’s Equation 268 9.3. Convergence of Continued Fractions 274 9.4. Continued Fraction Expansions of Real Numbers 280 9.5. Purely Periodic Continued Fractions 284 9.6. Continued Fractions of Irrational Quadratic Numbers 289 Continued Fractions—Review 296 Chapter 10. Class Groups of Positive Discriminant 300 10.1. Class Groups of Indefinite Quadratic Forms 300 10.2. Genera of Quadratic Forms and Ideals 306 10.3. Continued Fractions of Irrational Quadratic Numbers 311 10.4. Equivalence of Indefinite Quadratic Forms 316 Class Groups of Positive Discriminant—Review 319 Chapter 11. Representations by Indefinite Forms 322 11.1. The Continued Fraction of a Quadratic Form 322 11.2. Units and Automorphs 331 11.3. Existence of Representations by Indefinite Forms 335 11.4. Constructing Representations by Indefinite Forms 341 Representations by Indefinite Forms—Review 346 Part Five: Quadratic Recursive Sequences 348 Chapter 12. Properties of Recursive Sequences 350 12.1. Divisibility Properties of Quadratic RecursiveSequences 351 12.2. Periodicity of Quadratic Recursive Sequences 358 12.3. Suborder Functions 364 12.4. Suborder Sequences 371 Properties of Recursive Sequences—Review 379 Chapter 13. Applications of Quadratic Recursive Sequences 382 13.1. Recursive Sequences and Automorphs 382 13.2. An Application to Pell’s Equation 386 13.3. Quadratic Subdomains of Positive Discriminant 391 Applications of Quadratic Recursive Sequences—Review 396 Concluding Remarks 398 References and Suggested Reading 399 List of Notation 402 Index 406 Back Cover 410 Quadratic Number Theory is an introduction to algebraic number theory for readers with a moderate knowledge of elementary number theory and some familiarity with the terminology of abstract algebra. By restricting attention to questions about squares the author achieves the dual goals of making the presentation accessible to undergraduates and reflecting the historical roots of the subject. The representation of integers by quadratic forms is emphasized throughout the text. Lehman introduces an innovative notation for ideals of a quadratic domain that greatly facilitates computation and he uses this to particular effect. The text has an unusual focus on actual computation. This focus, and this notation, serve the author's historical purpose as well; ideals can be seen as number-like objects, as Kummer and Dedekind conceived of them. The notation can be adapted to quadratic forms and provides insight into the connection between quadratic forms and ideals. The computation of class groups and continued fraction representations are featured the author's notation makes these computations particularly illuminating. Quadratic Number Theory, with its exceptionally clear prose, hundreds of exercises, and historical motivation, would make an excellent textbook for a second undergraduate course in number theory. The clarity of the exposition would also make it a terrific choice for independent reading. It will be exceptionally useful as a fruitful launching pad for undergraduate research projects in algebraic number theory. Offers an introduction to algebraic number theory for readers with a moderate knowledge of elementary number theory and some familiarity with the terminology of abstract algebra. With its exceptionally clear prose, hundreds of exercises, and historical motivation, this is a textbook for a second undergraduate course in number theory.