Discovering abstract algebra:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, Rhode Island
MAA Press, an imprint of American Mathematical Society
[2021]
|
| Schriftenreihe: | AMS/MAA textbooks
Volume 67 |
| Schlagworte: |
General and overarching topics; collections
> Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematics in general
Field theory and polynomials
> Instructional exposition (textbooks, tutorial papers, etc.) pertaining to field theory
Commutative algebra
> Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra
Linear and multilinear algebra; matrix theory
> Instructional exposition (textbooks, tutorial papers, etc.)
|
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes index 2109 |
| Beschreibung: | xviii, 199 Seiten Diagramme |
| ISBN: | 9781470464424 |
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| 653 | 0 | |a Commutative algebra -- Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra | |
| 653 | 0 | |a Linear and multilinear algebra; matrix theory -- Instructional exposition (textbooks, tutorial papers, etc.) | |
| 653 | 0 | |a Group theory and generalizations -- Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra | |
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Datensatz im Suchindex
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|---|---|
| adam_text | Contents Acknowledgments xi To the instructor xiii To the student xvii Part 1 Group theory 1 Introduction 1.1 A brief backstory 1.2 Properties of the integers 2 Binary operations 2.1 Closure 2.2 Binary tables 2.3 Isomorphic structures 3 Groups 3.1 Basic properties of groups 3.2 Group notation 3.3 Group tables and the order of a group 4 Subgroups and generating sets 4.1 Subgroups 4.2 The center of a group 4.3 Generating sets 5 Applications of subgroups 5.1 Cosets 5.2 Lagrange’s theorem 5.3 Conjugation Part 2 Types of groups 6 Quotient groups 6.1 Homomorphisms and kernel 6.2 Normal subgroups 6.3 The natural projection homomorphism 1 3 3 4 7 7 10 11 15 15 18 20 23 23 26 27 31 31 33 34 37 39 39 41 44 vii
viii 7 Contents Cyclic groups 7.1 Properties of cyclic groups 7.2 Infinite cyclic groups 7.3 Finite cyclic groups 8 Direct products 8.1 External direct products 8.2 Finitely generated abelian groups 9 The isomorphism theorems 9.1 The first isomorphism theorem 9.2 Quotients of finitely generated abelian groups 9.3 The second and third isomorphism theorems 10 The symmetric groups 10.1 Permutations 10.2 Dihedral groups 10.3 Cayley’s theorem 11 Alternating groups 11.1 11.2 11.3 11.4 11.5 Orbits and cycles Transpositions and the parity of a permutation The alternating group Generating sets for symmetric groups The simplicity of ճ5 Part 3 Ring theory 12 Rings 12.1 Basic properties of rings 12.2 Homomorphisms 12.3 Polynomials 13 Commutative rings 13.1 Integral domains 13.2 The Ring Zn 13.3 Polynomials over integral domains 14 Fields 14.1 The field of quotients 14.2 The characteristic of a ring 14.3 Polynomials over a field 15 Quotient rings 15.1 Ideals 15.2 Ideals in commutative rings 15.3 Ideals in polynomial rings 47 47 48 49 53 53 55 59 59 60 62 65 65 66 68 71 71 73 74 75 78 81 83 83 87 89 93 93 95 96 99 99 101 102 105 105 106 108
Contents IX Part 4 Linear algebra 111 16 Vector spaces 113 113 116 117 119 16.1 16.2 16.3 16.4 Basic properties of vector spaces Linear combinations and span Linear independence and bases The dimension of a vector space 17 Linear transformations 17.1 17.2 17.3 17.4 17.5 Bases and linear transformations Rank and nullity Eigenvectors Linear operators Dual spaces Part 5 Field theory 18 Extension fields 18.1 Degree of an extension 18.2 Simple extensions 18.3 Splitting fields 123 123 125 126 127 129 133 135 135 136 139 19.1 Algebraic elements 19.2 Number fields 19.3 Finite fields 141 141 143 144 Part 6 Intermediate group theory 149 19 Algebraic extensions 20 Group actions 20.1 Definitions and examples 20.2 Orbits and stabilizers 20.3 Counting orbits 21 The Sylow theorems 21.1 21.2 21.3 21.4 The class equation The normalizer The Sylow theorems Applications to simple groups Appendices A Relations and functions A.1 Equivalence relations A.2 Functions A.3 Bijections and inverse functions В Matrices B.l Matrix algebra B.2 Matrix inverses 151 151 154 156 161 161 163 163 165 169 171 171 173 175 179 179 182
Contents B.З C Determinants Complex numbers C.l C.2 C.3 Index Complex arithmetic The geometry of complex numbers Complex solutions of equations 183 187 187 189 193 197
|
| adam_txt |
Contents Acknowledgments xi To the instructor xiii To the student xvii Part 1 Group theory 1 Introduction 1.1 A brief backstory 1.2 Properties of the integers 2 Binary operations 2.1 Closure 2.2 Binary tables 2.3 Isomorphic structures 3 Groups 3.1 Basic properties of groups 3.2 Group notation 3.3 Group tables and the order of a group 4 Subgroups and generating sets 4.1 Subgroups 4.2 The center of a group 4.3 Generating sets 5 Applications of subgroups 5.1 Cosets 5.2 Lagrange’s theorem 5.3 Conjugation Part 2 Types of groups 6 Quotient groups 6.1 Homomorphisms and kernel 6.2 Normal subgroups 6.3 The natural projection homomorphism 1 3 3 4 7 7 10 11 15 15 18 20 23 23 26 27 31 31 33 34 37 39 39 41 44 vii
viii 7 Contents Cyclic groups 7.1 Properties of cyclic groups 7.2 Infinite cyclic groups 7.3 Finite cyclic groups 8 Direct products 8.1 External direct products 8.2 Finitely generated abelian groups 9 The isomorphism theorems 9.1 The first isomorphism theorem 9.2 Quotients of finitely generated abelian groups 9.3 The second and third isomorphism theorems 10 The symmetric groups 10.1 Permutations 10.2 Dihedral groups 10.3 Cayley’s theorem 11 Alternating groups 11.1 11.2 11.3 11.4 11.5 Orbits and cycles Transpositions and the parity of a permutation The alternating group Generating sets for symmetric groups The simplicity of ճ5 Part 3 Ring theory 12 Rings 12.1 Basic properties of rings 12.2 Homomorphisms 12.3 Polynomials 13 Commutative rings 13.1 Integral domains 13.2 The Ring Zn 13.3 Polynomials over integral domains 14 Fields 14.1 The field of quotients 14.2 The characteristic of a ring 14.3 Polynomials over a field 15 Quotient rings 15.1 Ideals 15.2 Ideals in commutative rings 15.3 Ideals in polynomial rings 47 47 48 49 53 53 55 59 59 60 62 65 65 66 68 71 71 73 74 75 78 81 83 83 87 89 93 93 95 96 99 99 101 102 105 105 106 108
Contents IX Part 4 Linear algebra 111 16 Vector spaces 113 113 116 117 119 16.1 16.2 16.3 16.4 Basic properties of vector spaces Linear combinations and span Linear independence and bases The dimension of a vector space 17 Linear transformations 17.1 17.2 17.3 17.4 17.5 Bases and linear transformations Rank and nullity Eigenvectors Linear operators Dual spaces Part 5 Field theory 18 Extension fields 18.1 Degree of an extension 18.2 Simple extensions 18.3 Splitting fields 123 123 125 126 127 129 133 135 135 136 139 19.1 Algebraic elements 19.2 Number fields 19.3 Finite fields 141 141 143 144 Part 6 Intermediate group theory 149 19 Algebraic extensions 20 Group actions 20.1 Definitions and examples 20.2 Orbits and stabilizers 20.3 Counting orbits 21 The Sylow theorems 21.1 21.2 21.3 21.4 The class equation The normalizer The Sylow theorems Applications to simple groups Appendices A Relations and functions A.1 Equivalence relations A.2 Functions A.3 Bijections and inverse functions В Matrices B.l Matrix algebra B.2 Matrix inverses 151 151 154 156 161 161 163 163 165 169 171 171 173 175 179 179 182
Contents B.З C Determinants Complex numbers C.l C.2 C.3 Index Complex arithmetic The geometry of complex numbers Complex solutions of equations 183 187 187 189 193 197 |
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| isbn | 9781470464424 |
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| spelling | Osoinach, John K. 1968- Verfasser (DE-588)1252370229 aut Discovering abstract algebra John K. Osoinach, Jr Providence, Rhode Island MAA Press, an imprint of American Mathematical Society [2021] xviii, 199 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier AMS/MAA textbooks Volume 67 Includes index 2109 Universelle Algebra (DE-588)4061777-4 gnd rswk-swf Algebra, Abstract General and overarching topics; collections-- Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematics in general Field theory and polynomials -- Instructional exposition (textbooks, tutorial papers, etc.) pertaining to field theory Commutative algebra -- Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra Linear and multilinear algebra; matrix theory -- Instructional exposition (textbooks, tutorial papers, etc.) Group theory and generalizations -- Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra (DE-588)4123623-3 Lehrbuch gnd-content Universelle Algebra (DE-588)4061777-4 s DE-604 AMS/MAA textbooks Volume 67 (DE-604)BV045059717 Volume 67 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=032940848&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
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| title | Discovering abstract algebra |
| title_auth | Discovering abstract algebra |
| title_exact_search | Discovering abstract algebra |
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| title_full | Discovering abstract algebra John K. Osoinach, Jr |
| title_fullStr | Discovering abstract algebra John K. Osoinach, Jr |
| title_full_unstemmed | Discovering abstract algebra John K. Osoinach, Jr |
| title_short | Discovering abstract algebra |
| title_sort | discovering abstract algebra |
| topic | Universelle Algebra (DE-588)4061777-4 gnd |
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