Theory of commutative fields:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English Japanese |
| Published: |
Providence, RI
American Mathemat. Soc.
1993
|
| Series: | Translations of mathematical monographs
125 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Aus d. Japan. übers. |
| Physical Description: | XV, 249 S. |
| ISBN: | 0821845721 |
Staff View
MARC
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Record in the Search Index
| _version_ | 1804122746296205312 |
|---|---|
| adam_text | Contents
Preface to the English Edition xi
Preface to the New Japanese Edition xiii
Preface to the Original Japanese Edition xv
Chapter 0. Prerequisites from Set Theory 1
§0. Basic symbols 1
§1. Mappings 1
§2. Ordered sets 3
§3. Partitions and equivalence relations 4
Chapter I. Groups, Rings, and Fields 7
§1. Groups 7
§2. Normal subgroups and homomorphisms 10
§3. Rings and fields 16
§4. Integral domains and prime ideals 20
§5. Polynomial rings 23
§6. Unique factorization 25
§7. Modules 29
§8. Symmetric forms and alternating forms 34
Exercises 37
Chapter II. Algebraic Extensions of Finite Degrees 43
§1. Basic notions 43
§2. Splitting fields 47
§3. Separability and inseparability 48
§4. Multiplicative groups of finite fields 51
§5. Simple extensions 51
§6. Normal extensions 54
§7. Invariants of a finite group 55
§8. The fundamental theorem of Galois 58
§9. Roots of unity and cyclic extensions 59
§ 10. Solvability of algebraic equations 63
§11. Construction problems 67
vii
viii CONTENTS
§ 12. Algebraically closed fields 71
Appendix 1 73
Appendix 2 74
Exercises 75
Chapter III. Transcendental Extensions 81
§ 1. Transcendence bases 81
§2. Tensor products over a field 83
§3. Derivations 87
§4. Separable extensions 91
§5. Regular extensions 94
§6. Noetherian rings 96
§7. Integral extensions and prime ideals 102
§8. The normalization theorem for polynomial rings 107
§9. Integral closures 111
§ 10. Algebraic varieties 113
§11. The C. conditions 118
§ 12. The theorem of Luroth 122
Appendix. A theorem on valuation rings and its applications 124
Exercises 126
Chapter IV. Valuations 135
§ 1. Multiplicative valuations 135
§2. Valuations of the rational number field 138
§3. Topology 140
§4. Topological groups and topological fields 145
§5. Completions 149
§6. Archimedean valuations and absolute values 152
§7. Additive valuations and valuation rings 154
§8. Approximation theorems 159
§9. Prolongations of a valuation 162
§ 10. The product formula 168
§11. Hensel s lemma 170
Exercises 182
Chapter V. Formally Real Fields 191
§ 1. Ordered fields, formally real fields, and real closed fields 191
§2. Real closures 196
§3. Hilbert s 17th Problem 202
§4. A valuation corresponding to an order 206
Exercises 211
CONTENTS ix
Chapter VI. Galois Theory of Algebraic Extensions of Infinite Degree 215
§ 1. Topology on a Galois group 215
§2. The fundamental theorem of Galois 217
§3. Splitting fields, inertia fields, and ramification fields 219
§4. Algebraic equaitons of high degrees 221
Exercises 224
Answers and Hints 227
Index of Symbols 243
Subject Index 245
|
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| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.74 |
| dewey-search | 512/.74 |
| dewey-sort | 3512 274 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| id | DE-604.BV008356144 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:18:46Z |
| institution | BVB |
| isbn | 0821845721 |
| language | English Japanese |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-005517486 |
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| owner | DE-12 DE-384 DE-703 DE-11 DE-20 |
| owner_facet | DE-12 DE-384 DE-703 DE-11 DE-20 |
| physical | XV, 249 S. |
| publishDate | 1993 |
| publishDateSearch | 1993 |
| publishDateSort | 1993 |
| publisher | American Mathemat. Soc. |
| record_format | marc |
| series | Translations of mathematical monographs |
| series2 | Translations of mathematical monographs |
| spelling | Nagata, Masayoshi 1927-2008 Verfasser (DE-588)13883069X aut Kakantairon Theory of commutative fields Masayoshi Nagata Providence, RI American Mathemat. Soc. 1993 XV, 249 S. txt rdacontent n rdamedia nc rdacarrier Translations of mathematical monographs 125 Aus d. Japan. übers. Commutatieve algebra's gtt Corps algébriques Corps algébriques ram Corps commutatif Corps formellement réel Extension galoisienne infinie Loi commutative (Mathématiques) Problème Hilbert 17 Veldentheorie gtt Algebraic fields Commutative law (Mathematics) Körper Algebra (DE-588)4308063-7 gnd rswk-swf Körper Algebra (DE-588)4308063-7 s DE-604 Translations of mathematical monographs 125 (DE-604)BV000002394 125 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005517486&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Nagata, Masayoshi 1927-2008 Theory of commutative fields Translations of mathematical monographs Commutatieve algebra's gtt Corps algébriques Corps algébriques ram Corps commutatif Corps formellement réel Extension galoisienne infinie Loi commutative (Mathématiques) Problème Hilbert 17 Veldentheorie gtt Algebraic fields Commutative law (Mathematics) Körper Algebra (DE-588)4308063-7 gnd |
| subject_GND | (DE-588)4308063-7 |
| title | Theory of commutative fields |
| title_alt | Kakantairon |
| title_auth | Theory of commutative fields |
| title_exact_search | Theory of commutative fields |
| title_full | Theory of commutative fields Masayoshi Nagata |
| title_fullStr | Theory of commutative fields Masayoshi Nagata |
| title_full_unstemmed | Theory of commutative fields Masayoshi Nagata |
| title_short | Theory of commutative fields |
| title_sort | theory of commutative fields |
| topic | Commutatieve algebra's gtt Corps algébriques Corps algébriques ram Corps commutatif Corps formellement réel Extension galoisienne infinie Loi commutative (Mathématiques) Problème Hilbert 17 Veldentheorie gtt Algebraic fields Commutative law (Mathematics) Körper Algebra (DE-588)4308063-7 gnd |
| topic_facet | Commutatieve algebra's Corps algébriques Corps commutatif Corps formellement réel Extension galoisienne infinie Loi commutative (Mathématiques) Problème Hilbert 17 Veldentheorie Algebraic fields Commutative law (Mathematics) Körper Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005517486&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000002394 |
| work_keys_str_mv | AT nagatamasayoshi kakantairon AT nagatamasayoshi theoryofcommutativefields |