Number theory and geometry: an introduction to arithmetic geometry
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
[2019]
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| Series: | Pure and applied undergraduate texts
35 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | xv, 488 Seiten Illustrationen, Diagramme |
| ISBN: | 9781470450168 |
Staff View
MARC
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Record in the Search Index
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| adam_text | CONTENTS Preface xiii Chapter 1. Introduction 1.1. Roots of Polynomials 1.2. Lines 1.3. Quadratic Equations and Conic Sections 1.4. Cubic Equations and Elliptic Curves 1.5. Curves of Higher Degree 1.6. Diophantine Equations 1.7. Hilbert’s Tenth Problem 1.8. Exercises 1 2 5 7 11 14 16 21 23 Part 1. Integers, Polynomials, Lines, and Congruences Chapter 2. The Integers 2.1. The Axioms of Z 2.2. Consequences of the Axioms 2.3. The Principle of Mathematical Induction 2.4. The Division Theorem 2.5. The Greatest Common Divisor 2.6. Euclid’s Algorithm to Calculate a GCD 2.7. Bezout’s Identity 2.8. Integral and Rational Roots of Polynomials 2.9. Integral and Rational Points in a Line 2.10. The Fundamental Theorem of Arithmetic 2.11. Exercises vii 29 29 31 33 38 41 42 43 47 48 51 55
viii Contents Chapter 3. The Prime Numbers 61 3.1. The Sieve of Eratosthenes 62 3.2. The Infinitude of the Primes 63 3.3. Theorems on the Distributionof Primes 67 3.4. Famous Conjectures about Prime Numbers 72 3.5. Exercises 79 Chapter 4. Congruences 83 4.1. The Definition of Congruence 84 4.2. Basic Properties of Congruences 86 4.3. Cancellation Properties of Congruences 89 4.4. Linear Congruences 90 4.5. Systems of Linear Congruences 94 4.6. Applications 102 4.7. Exercises 113 Chapter 5. Groups, Rings, and Fields 119 5.1. Z/mZ 119 5.2. Groups 124 5.3. Rings 130 5.4. Fields 138 5.5. Rings of Polynomials 140 5.6. Exercises 149 Chapter 6. Finite Fields 155 6.1. An Example 155 6.2. Polynomial Congruences 156 6.3. Irreducible Polynomials 159 6.4. Fields with pn Elements 160 6.5. 6.6. Fields with p2 Elements Fields with s Elements 161 163 6.7. Exercises 164 Chapter 7. The Theorems of Wilson,Fermat, and Euler 167 7.1. Wilson’s Theorem 167 7.2. Fermat’s (Little) Theorem 170 7.3. Euler’s Theorem 176 7.4. Euler’s Phi Function 181 7.5. Applications 184 7.6. Exercises 188
Contents Chapter 8. ix Primitive Roots 193 8.1. 8.2. Multiplicative Order Primitive Roots 195 200 8.3. 8.4. 8.5. Universal Exponents Existence of Primitive Roots Modulo p Primitive Roots Modulo pk 203 205 210 8.6. 8.7. Indices Existence of Primitive Roots Modulo m 214 220 8.8. 8.9. The Structure of (Z/pfeZ)x Applications 222 224 8.10. Exercises 230 Part 2. Quadratic Congruences andQuadratic Equations Chapter 9. An Introduction to Quadratic Equations 9.1. Product of Two Lines 237 238 9.2. 9.3. A Classification: Parabolas, Ellipses, and Hyperbolas Rational Parametrizations of Conics 248 255 9.4. 9.5. Integral Points on Quadratic Equations Exercises 260 268 Chapter 10. Quadratic Congruences 10.1. The Quadratic Formula 271 272 10.2. 10.3. 10.4. 10.5. Quadratic Residues The Legendre Symbol The Law of Quadratic Reciprocity The Jacobi Symbol 275 279 284 290 10.6. 10.7. 10.8. Cipolla’s Algorithm Applications Exercises 296 298 305 Chapter 11. The Hasse-Minkowski Theorem 309 11.1. 11.2. 11.3. 11.4. 11.5. Quadratic Forms The Hasse-Minkowski Theorem An Example of Hasse-Minkowski Polynomial Congruences for Prime Powers The p-Adic Numbers 309 313 318 324 328 11.6. 11.7. Hensel’s Lemma Exercises 331 333
Contents x Chapter 12. Circles, Ellipses, and the Sum of Two Squares Problem 337 12.1. 12.2. Rational and Integral Points on a Circle Pythagorean Triples 337 343 12.3. Fermat’s Last Theorem for n = 4 347 12.4. Ellipses 348 12.5. Quadratic Fields and Norms 350 12.6. Integral Points on Ellipses 353 12.7. Primes of the Form X2 + BY2 353 12.8. Exercises 356 Chapter 13. Continued Fractions 13.1. Finite Continued Fractions 361 363 13.2. 13.3. Infinite Continued Fractions Approximations of Irrational Numbers 370 386 13.4. Exercises 389 Chapter 14. Hyperbolas and Pell’s Equation 393 14.1. 14.2. Square Hyperbolas Pell’s Equation x2 — By2 = 1 393 395 14.3. Generalized Pell’s Equations x2 — By2 = N 401 14.4. Exercises 409 Part 3. Cubic Equations and Elliptic Curves Chapter 15. An Introduction to CubicEquations 413 15.1. 15.2. The Projective Line and Projective Space Singular Cubic Curves 415 422 15.3. Weierstrass Equations 425 15.4. Exercises 433 Chapter 16. Elliptic Curves 437 16.1. 16.2. Definition Integral Points 438 441 16.3. 16.4. The Group Structure on E(Q) The Torsion Subgroup 441 447 16.5. 16.6. Elliptic Curves over Finite Fields The Rank and the Free Part of E(Q) 449 455 16.7. 16.8. Descent and the Weak Mordell-Weil Theorem Homogeneous Spaces 459 467 16.9. Application: The Elliptic Curve Diffie-Hellman Key Exchange 471 16.10. Exercises 473
Contents Xl Bibliography 479 Index 483
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| dewey-full | 512.7/4 |
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| dewey-ones | 512 - Algebra |
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| dewey-search | 512.7/4 |
| dewey-sort | 3512.7 14 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| indexdate | 2024-07-10T08:29:16Z |
| institution | BVB |
| isbn | 9781470450168 |
| language | English |
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| physical | xv, 488 Seiten Illustrationen, Diagramme |
| publishDate | 2019 |
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| publishDateSort | 2019 |
| publisher | American Mathematical Society |
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| series | Pure and applied undergraduate texts |
| series2 | Pure and applied undergraduate texts The Sally series |
| spelling | Lozano-Robledo, Álvaro 1978- Verfasser (DE-588)143876678 aut Number theory and geometry an introduction to arithmetic geometry Álvaro Lozano-Robledo Providence, Rhode Island American Mathematical Society [2019] xv, 488 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Pure and applied undergraduate texts 35 The Sally series Includes bibliographical references and index Arithmetische Geometrie (DE-588)4131383-5 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Arithmetical algebraic geometry / Textbooks Number theory / Textbooks Arithmetische Geometrie (DE-588)4131383-5 s Zahlentheorie (DE-588)4067277-3 s DE-604 9781470451905 Pure and applied undergraduate texts 35 (DE-604)BV035489189 35 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031264107&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lozano-Robledo, Álvaro 1978- Number theory and geometry an introduction to arithmetic geometry Pure and applied undergraduate texts Arithmetische Geometrie (DE-588)4131383-5 gnd Zahlentheorie (DE-588)4067277-3 gnd |
| subject_GND | (DE-588)4131383-5 (DE-588)4067277-3 |
| title | Number theory and geometry an introduction to arithmetic geometry |
| title_auth | Number theory and geometry an introduction to arithmetic geometry |
| title_exact_search | Number theory and geometry an introduction to arithmetic geometry |
| title_full | Number theory and geometry an introduction to arithmetic geometry Álvaro Lozano-Robledo |
| title_fullStr | Number theory and geometry an introduction to arithmetic geometry Álvaro Lozano-Robledo |
| title_full_unstemmed | Number theory and geometry an introduction to arithmetic geometry Álvaro Lozano-Robledo |
| title_short | Number theory and geometry |
| title_sort | number theory and geometry an introduction to arithmetic geometry |
| title_sub | an introduction to arithmetic geometry |
| topic | Arithmetische Geometrie (DE-588)4131383-5 gnd Zahlentheorie (DE-588)4067277-3 gnd |
| topic_facet | Arithmetische Geometrie Zahlentheorie |
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