Introduction to quadratic forms:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
1971
|
| Ausgabe: | 2. print., corr. |
| Schriftenreihe: | Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen
117 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | 342 S. Ill. |
| ISBN: | 3540029842 0387029842 |
Internformat
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Datensatz im Suchindex
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| adam_text | Titel: Introduction to quadratic forms
Autor: O Meara, Onorato T
Jahr: 1971
Contents Prerequisites and Notation.......................XI Part One Arithmetic Theory oi Fields Chapter I. Valuated Fields...................... 1 11. Valuations.......................... 1 12. Archimedean valuations.................... 14 13. Non-archimedean valuations..................20 14. Prolongation of a complete valuation to a finite extension.....28 15. Prolongation of any valuation to a finite separable extension .... 30 16. Discrete valuations......................37 Chapter II. Dedekind Theory of Ideals.................41 21. Dedekind axioms for S.....................42 22. Ideal theory.........................44 23. Extension fields........................52 Chapter III. Fields of Number Theory.................54 31. Rational global fields .....................54 32. Local fields..........................59 33. Global fields.........................65 Part Two Abstract Theory of Quadratic Forms Chapter IV. Quadratic Forms and the Orthogonal Group.........82 41. Forms, matrices and spaces...................82 42. Quadratic spaces.......................88 43. Special subgroups of O n (V) ................... 100 Chapter V. The Algebras of Quadratic Forms..............112 51. Tensor products........................113 52. Wedderburn’s theorem on central simple algebras.........118 53. Extending the field of scalars..................129 54. The Clifford algebra......................131 55. The spinor norm .......................137 56. Special subgroups of 0„ (F)...................141 57. Quaternion algebras......................142 58. The Hasse algebra.......................149
X Contents Part Three Arithmetic Theory of Quadratic Forms over Fields Chapter VI. The Equivalence of Quadratic Forms............154 61. Complete archimedean fields..................154 62. Finite fields..........................157 63. Local fields..........................158 64. Global notation........................172 65. Squares and norms in global fields................173 66. Quadratic forms over global fields................186 Chapter VII. Hilbert’s Reciprocity Law.................190 71. Proof of the reciprocity law...................190 72. Existence of forms with prescribed local behavior.........203 73. The quadratic reciprocity law..................205 Part Four Arithmetic Theory of Quadratic Forms over Rings Chapter VIII. Quadratic Forms over Dedekind Domains.........208 81. Abstract lattices........................208 82. Lattices in quadratic spaces...................220 Chapter IX. Integral Theory of Quadratic Forms over Local Fields.....239 91. Generalities..........................239 92. Classification of lattices over non-dyadic fields...........246 93. Classification of lattices over dyadic fields ............250 94. Effective determination of the invariants.............279 95. Special subgroups of O n (V) ...................280 Chapter X. Integral Theory of Quadratic Forms over Global Fields.....284 101. Elementary properties of the orthogonal group over arithmetic fields 285 102. The genus and the spinor genus.................297 103. Finiteness of class number...................305 104. The class and the spinor genus in the indefinite case........311 105. The indecomposable splitting of a definite lattice.........321 106. Definite unimodular lattices over the rational integers.......323 Bibliography.............................336 Index................................337
|
| adam_txt |
Titel: Introduction to quadratic forms
Autor: O'Meara, Onorato T
Jahr: 1971
Contents Prerequisites and Notation.XI Part One Arithmetic Theory oi Fields Chapter I. Valuated Fields. 1 11. Valuations. 1 12. Archimedean valuations. 14 13. Non-archimedean valuations.20 14. Prolongation of a complete valuation to a finite extension.28 15. Prolongation of any valuation to a finite separable extension . 30 16. Discrete valuations.37 Chapter II. Dedekind Theory of Ideals.41 21. Dedekind axioms for S.42 22. Ideal theory.44 23. Extension fields.52 Chapter III. Fields of Number Theory.54 31. Rational global fields .54 32. Local fields.59 33. Global fields.65 Part Two Abstract Theory of Quadratic Forms Chapter IV. Quadratic Forms and the Orthogonal Group.82 41. Forms, matrices and spaces.82 42. Quadratic spaces.88 43. Special subgroups of O n (V) . 100 Chapter V. The Algebras of Quadratic Forms.112 51. Tensor products.113 52. Wedderburn’s theorem on central simple algebras.118 53. Extending the field of scalars.129 54. The Clifford algebra.131 55. The spinor norm .137 56. Special subgroups of 0„ (F).141 57. Quaternion algebras.142 58. The Hasse algebra.149
X Contents Part Three Arithmetic Theory of Quadratic Forms over Fields Chapter VI. The Equivalence of Quadratic Forms.154 61. Complete archimedean fields.154 62. Finite fields.157 63. Local fields.158 64. Global notation.172 65. Squares and norms in global fields.173 66. Quadratic forms over global fields.186 Chapter VII. Hilbert’s Reciprocity Law.190 71. Proof of the reciprocity law.190 72. Existence of forms with prescribed local behavior.203 73. The quadratic reciprocity law.205 Part Four Arithmetic Theory of Quadratic Forms over Rings Chapter VIII. Quadratic Forms over Dedekind Domains.208 81. Abstract lattices.208 82. Lattices in quadratic spaces.220 Chapter IX. Integral Theory of Quadratic Forms over Local Fields.239 91. Generalities.239 92. Classification of lattices over non-dyadic fields.246 93. Classification of lattices over dyadic fields .250 94. Effective determination of the invariants.279 95. Special subgroups of O n (V) .280 Chapter X. Integral Theory of Quadratic Forms over Global Fields.284 101. Elementary properties of the orthogonal group over arithmetic fields 285 102. The genus and the spinor genus.297 103. Finiteness of class number.305 104. The class and the spinor genus in the indefinite case.311 105. The indecomposable splitting of a definite lattice.321 106. Definite unimodular lattices over the rational integers.323 Bibliography.336 Index.337 |
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| index_date | 2024-07-02T16:16:06Z |
| indexdate | 2024-07-09T20:50:54Z |
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| isbn | 3540029842 0387029842 |
| language | English |
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| series | Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen |
| series2 | Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen |
| spelling | O'Meara, Onorato T. 1928-2018 Verfasser (DE-588)121509559 aut Introduction to quadratic forms O. T. O'Meara 2. print., corr. Berlin [u.a.] Springer 1971 342 S. Ill. txt rdacontent n rdamedia nc rdacarrier Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 117 Formes quadratiques Formes quadratiques ram Forms, Quadratic Quadratische Form (DE-588)4128297-8 gnd rswk-swf 1\p (DE-588)4151278-9 Einführung gnd-content Quadratische Form (DE-588)4128297-8 s DE-604 Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 117 (DE-604)BV000000395 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015333225&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | O'Meara, Onorato T. 1928-2018 Introduction to quadratic forms Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen Formes quadratiques Formes quadratiques ram Forms, Quadratic Quadratische Form (DE-588)4128297-8 gnd |
| subject_GND | (DE-588)4128297-8 (DE-588)4151278-9 |
| title | Introduction to quadratic forms |
| title_auth | Introduction to quadratic forms |
| title_exact_search | Introduction to quadratic forms |
| title_exact_search_txtP | Introduction to quadratic forms |
| title_full | Introduction to quadratic forms O. T. O'Meara |
| title_fullStr | Introduction to quadratic forms O. T. O'Meara |
| title_full_unstemmed | Introduction to quadratic forms O. T. O'Meara |
| title_short | Introduction to quadratic forms |
| title_sort | introduction to quadratic forms |
| topic | Formes quadratiques Formes quadratiques ram Forms, Quadratic Quadratische Form (DE-588)4128297-8 gnd |
| topic_facet | Formes quadratiques Forms, Quadratic Quadratische Form Einführung |
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