Algebra in action: a course in groups, rings, and fields
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
[2017]
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| Series: | Pure and applied undergraduate texts
27 : The Sally series |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xvii, 675 Seiten Illustrationen |
| ISBN: | 9781470428495 |
Staff View
MARC
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Record in the Search Index
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| adam_text | Contents Preface xiii Part 1. (Mostly Finite) Group Theory Chapter 1. Four Basic Examples 1.1. Symmetries of a Square 3 4 1.2. 1.3. 1-1 and Onto Functions Integers mod n and Elementary Properties of Integers 9 20 1.4. 1.5. Invertible Matrices More Problems and Projects 29 33 Chapter 2. Groups: The Basics 2.1. Definitions and Examples 2.2. 2.3. 37 37 Cancellation Properties Cyclic Groups and the Order of an Element 42 45 2.4. Isomorphisms 2.5. Direct Products (New Groups from Old Groups) 2.6. Subgroups 2.7. More Problems and Projects 53 58 61 69 Chapter 3. The Alternating Groups 3.1. Permutations, Cycles, and Transpositions 3.2. Even and Odd Permutations and An 3.3. More Problems and Projects Chapter 4. Group Actions 4.1. Definition and Examples 75 75 78 82 85 85 vii
Contents viii 4.2. The Cayley Graph of a Group Action* 4.3. Stabilizers 4.4. Orbits 4.5. More Problems and Projects 91 93 96 105 Chapter 5. A Subgroup Acts on the Group: Cosets and Lagrange’s Theorem 109 5.1. Translation Action and Cosets 109 5.2. Lagrange’s Theorem 115 5.3. Application to Number Theory* 120 5.4. More Problems and Projects 123 Chapter 6. A Group Acts on Itself: Counting and the Conjugation Action 6.1. The Fundamental Counting Principle і 6.2. The Conjugation Action 6.3. The Class Equation and Groups of Order p2 6.4. More Problems and Projects 129 129 133 138 141 Chapter 7. Acting on Subsets, Cosets, and Subgroups: The Sylow Theorems 143 7.1. Binomial Coefficients mod p 144 7.2. The Sylow E(xistence) Theorem 147 7.3. The Number and Conjugacy of Sylow Subgroups* 149 Chapter 8. Counting the Number of Orbits* 8.1. The Cauchy-Frobenius Counting Lemma 8.2. Combinatorial Applications of the Counting Lemma 8.3. More Problems and Projects 155 155 158 163 Chapter 9. The Lattice of Subgroups* 9.1. Partially Ordered Sets, Hasse Diagrams, and Lattices 9.2. Edge Lengths and Partial Lattice Diagrams 9.3. More Problems and Projects 167 167 174 183 Chapter 10. 10.1. 10.2. 10.3. 10.4. Chapter 11.1. 11.2. 11.3. Acting on Its Subgroups: Normal Subgroups and Quotient Groups Normal Subgroups The Normalizer Quotient Groups More Problems and Projects ՛; ; , 187 187 193 197 205 11. Group Homomorphisms Definitions, Examples, and ElementaryProperties The Kernel and the Image Homomorphisms, Normal Subgroups,andQuotient Groups 209 210 214 217
Contents 11.4. 11.5. 11.6. 11.7. ix Actions and Homomorphisms The Homomorphism Theorems Automorphisms and Inner-automorphisms* More Problems and Projects Chapter 12. 223 227 238 243 Using Sylow Theorems to Analyze Finite Groups* 249 12.1. p-groups · ■ ՛■■■■. 12.2. Acting on Cosets and Existence of Normal Subgroups 12.3. Applying the Sylow Theorems 12.4. As Is the Only Simple Group of Order 60 249 252 254 262 Chapter 13.1. 13.2. 13.3. 13.4. 13. Direct and Semidirect Products* Direct Products of Groups Fundamental Theorem of Finite Abelian Groups Semidirect Products Groups of Very Small Order 269 270 273 277 282 Chapter 14.1. 14.2. 14.3. 14. Solvable and Nilpotent Groups* Solvable Groups Nilpotent Groups The Jordan-Hölder Theorem 285 285 294 300 Part 2. (Mostly Commutative) Ring Theory Chapter 15.1. 15.2. 15.3. 15. Rings Diophantine Equations and Rings Rings, Integral Domains, Division Rings, and Fields Finite Integral Domains 309 310 316 325 Chapter 16.1. 16.2. 16.3. 16.4. 16. Homomorphisms, Ideals, and Quotient Rings 327 Subrings, Homomorphisms, and Ideals ՛ 327 Quotient Rings and Homomorphism Theorems 337 Characteristic of Rings with Identity, Integral Domains, and Fields 342 Manipulating Ideals* 346 Chapter 17. Field of Fractions and Localization 17.1. Field of Fractions and Localization of an Integral Domain 17.2. Localization of Commutative Rings with Identity* 353 354 360 Chapter 18.1. 18.2. 18.3. 367 367 379 382 18. Factorization, EDs, PIDs, and UFDs Factorization in Commutative Rings Ascending Chain Condition and Noetherian Rings A PID is a UFD
Contents x 18.4. Euclidean Domains 18.5. The Greatest Common Divisor* 391 18.6. More Problems and Projects 396 Chapter 19. . 387 Polynomial Rings 403 19.1. Polynomials : 403 19.2. 19.3. К a field = K[x] an ED Roots of Polynomials and Construction of Finite Fields 407 410 19.4. 19.5. R UFD R[æ] UFD and Gauss’s Lemma Irreducibility Criteria ľ 416 425 19.6. 19.7. Hilbert Basis Theorem* More Problems and Projects 428 430 , Chapter 20. Gaussian Integers and (a little) Number Theory* . 20.1. Gaussian Integers 20.2. 435 437 Unique Factorization and Diophantine Equations 444 Part 3. Fields and Galois Theory Chapter 21. Introducing Field Theory and Galois Theory 451 21.1. 21.2. The Classical Problems of Field Theory Roots of Equations, Fields, and Groups—An Example 451 454 21.3. A Quick Review of Ring Theory 456 Chapter 22. ֊ Field Extensions 459 22.1. 22.2. Simple and Algebraic Extensions A Quick Review of Vector Spaces 459 465 22.3. The Degree of an Extension 467 Chapter 23. 23.1. 23.2. Straightedge and Compass Constructions 477 The Field of Constructible Numbers Characterizing Constructible Numbers 478 484 Chapter 24. Splitting Fields and Galois Groups 24.1. Roots of Polynomials, Field Extensions, and E-isomorphisms 24.2. 24.3. Splitting Fields Galois Groups and Their Actions on Roots , Chapter 25. Galois, Normal, and Separable Extensions 25.1. Subgroups of the Galois Group and IntermediateFields : 25.2. 25.3. 25.4. Galois, Normal, and Separable Extensions More on Normal Extensions More on Separable Extensions 491 491 , 498 508 515 515 518 525 528
Contents 25.5. 25.6. xi Simple Extensions More Problems and Projects 532 539 Chapter 26.1. 26.2. 26.3. 26. Fundamental Theorem of Galois Theory Galois Groups and Fixed Fields Fundamental Theorem of GaloisTheory Examples of Galois Groups 547 547 551 558 Chapter 27.1. 27.2. 27.3. 27.4. 27. Finite Fields and CyclotomieExtensions Finite Fields Cyclotomie Extensions The Polynomial xn — a More Problems and Projects 565 565 572 580 585 Chapter 28.1. 28.2. 28.3. 28.4. 28. Radical Extensions, Solvable Groups, and the Quintic Solvability by Radicals A Solvable Polynomial Has a SolvableGalois Group A Solvable Galois Group Correspondsto a Solvable Polynomial* More Problems and Projects Appendix A. Hints for Selected Problems 589 589 594 598 606 611 Appendix B.Short Answers for Selected Problems 619 Appendix C. 623 Complete Solutions for Selected (Odd-Numbered) Problems Bibliography 661 Index 665
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| discipline | Mathematik |
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| id | DE-604.BV044305665 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T07:49:17Z |
| institution | BVB |
| isbn | 9781470428495 |
| language | English |
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| physical | xvii, 675 Seiten Illustrationen |
| publishDate | 2017 |
| publishDateSearch | 2017 |
| publishDateSort | 2017 |
| publisher | American Mathematical Society |
| record_format | marc |
| series | Pure and applied undergraduate texts |
| series2 | Pure and applied undergraduate texts The Sally series |
| spelling | Shahriari, Shahriar 1956- Verfasser (DE-588)141856734 aut Algebra in action a course in groups, rings, and fields Shahriar Shahriari Providence, Rhode Island American Mathematical Society [2017] © 2017 xvii, 675 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Pure and applied undergraduate texts 27 The Sally series Gruppentheorie (DE-588)4072157-7 gnd rswk-swf Algebra (DE-588)4001156-2 gnd rswk-swf Feldtheorie (DE-588)4016698-3 gnd rswk-swf Algebra (DE-588)4001156-2 s DE-604 Gruppentheorie (DE-588)4072157-7 s Feldtheorie (DE-588)4016698-3 s Pure and applied undergraduate texts 27 : The Sally series (DE-604)BV035489189 27 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=029709436&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Shahriari, Shahriar 1956- Algebra in action a course in groups, rings, and fields Pure and applied undergraduate texts Gruppentheorie (DE-588)4072157-7 gnd Algebra (DE-588)4001156-2 gnd Feldtheorie (DE-588)4016698-3 gnd |
| subject_GND | (DE-588)4072157-7 (DE-588)4001156-2 (DE-588)4016698-3 |
| title | Algebra in action a course in groups, rings, and fields |
| title_auth | Algebra in action a course in groups, rings, and fields |
| title_exact_search | Algebra in action a course in groups, rings, and fields |
| title_full | Algebra in action a course in groups, rings, and fields Shahriar Shahriari |
| title_fullStr | Algebra in action a course in groups, rings, and fields Shahriar Shahriari |
| title_full_unstemmed | Algebra in action a course in groups, rings, and fields Shahriar Shahriari |
| title_short | Algebra in action |
| title_sort | algebra in action a course in groups rings and fields |
| title_sub | a course in groups, rings, and fields |
| topic | Gruppentheorie (DE-588)4072157-7 gnd Algebra (DE-588)4001156-2 gnd Feldtheorie (DE-588)4016698-3 gnd |
| topic_facet | Gruppentheorie Algebra Feldtheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029709436&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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