Basic quadratic forms:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, R.I.
American Mathematical Society
2008
|
| Series: | Graduate studies in mathematics
90 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 255 S. |
| ISBN: | 9780821844656 0821844652 |
Staff View
MARC
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| 245 | 1 | 0 | |a Basic quadratic forms |c Larry J. Gerstein |
| 264 | 1 | |a Providence, R.I. |b American Mathematical Society |c 2008 | |
| 300 | |a XIII, 255 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Graduate studies in mathematics |v 90 | |
| 650 | 4 | |a Formes quadratiques | |
| 650 | 4 | |a Théorie des nombres | |
| 650 | 4 | |a Équations du second degré | |
| 650 | 4 | |a Forms, Quadratic | |
| 650 | 4 | |a Equations, Quadratic | |
| 650 | 4 | |a Number theory | |
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Record in the Search Index
| _version_ | 1804137747312541696 |
|---|---|
| adam_text | Contents
Preface
xi
Chapter
1.
A Brief Classical Introduction
1
§1.1.
Quadratic Forms as Polynomials
1
§1.2.
Representation and Equivalence; Matrix Connections;
Discriminants
4
Exercises
7
§1.3.
A Brief Historical Sketch, and Some References to the
Literature
7
Chapter
2.
Quadratic Spaces and Lattices
13
§2.1.
Fundamental Definitions
13
§2.2.
Orthogonal Splitting: Examples of Isometry and Non-isometry
16
Exercises
20
§2.3.
Representation. Splitting, and Isotropy; Invariants u(F) and
s(F)
21
§2.4.
The Orthogonal Group of a Space
26
§2.5.
Witt s Cancellation Theorem and Its Consequences
29
§2.6.
Witt s Chain Equivalence Theorem
34
§2.7.
Tensor Products of Quadratic Spaces; the Witt ring of a field
35
Exercises
39
§2.8.
Quadratic Spaces over Finite Fields
40
§2.9.
Hermitian Spaces
44
Exercises
49
vii
viii Contents
Chapter
3.
Valuations, Local Fields, and p-adic Numbers
51
§3.1.
Introduction to Valuations
51
§3.2.
Equivalence of Valuations; Prime Spots on a Field
54
Exercises
58
§3.3.
Completions, Qp, Residue Class Fields
59
§3.4.
Discrete Valuations
63
§3.5.
The Canonical Power Series Representation
64
§3.6.
Hensel s Lemma, the Local Square Theorem, and Local Fields
69
§3.7.
The Legendre Symbol; Recognizing Squares in Qp
74
Exercises
76
Chapter
4.
Quadratic Spaces over Qp
81
§4.1.
The Hubert Symbol
81
§4.2.
The
Hasse
Symbol (and an Alternative)
86
§4.3.
Classification of Quadratic Qp-Spaces
87
§4.4.
Hermitian Spaces over Quadratic Extensions of Qp
92
Exercises
94
Chapter
5.
Quadratic Spaces over
Q
97
§5.1.
The Product Formula and Hubert s Reciprocity Law
97
§5.2.
Extension of the Scalar Field
98
§5.3.
Local to Global: The Hasse-Minkowski Theorem
99
§5.4.
The Bruck-Ryser Theorem on Finite
Projective
Planes
105
§5.5.
Sums of Integer Squares (First Version)
109
Exercises 111
Chapter
6.
Lattices over Principal Ideal Domains
113
§6.1.
Lattice Basics
114
§6.2.
Valuations and Fractional Ideals
116
§6.3.
Invariant factors
118
§6.4.
Lattices on Quadratic Spaces
122
§6.5.
Orthogonal Splitting and Triple Diagonalization
124
§6.6.
The Dual of a Lattice
128
Exercises
130
§6.7.
Modular Lattices
133
§6.8.
Maximal Lattices
136
§6.9.
Unimodular Lattices and Pythagorean Triples
138
Contents ix
§6.10.
Remarks on Lattices over More General Rings
141
Exercises
142
Chapter
7.
Initial Integral Results
145
§7.1.
The Minimum of a Lattice; Definite Binary Z-Lattices
146
§7.2.
Hermite s Bound on
min L,
with a Supplement for fc[a;]-Lattices
149
§7.3.
Djokovic s Reduction of fcJxj-Lattices; Harder s Theorem
153
§7.4.
Finiteness of Class Numbers (The
Anisotropie
Case)
156
Exercises
158
Chapter
8.
Local Classification of Lattices
161
§8.1.
Jordan Splittings
161
§8.2.
Nondyadic Classification
164
§8.3.
Towards 2-adic Classification
165
Exercises
171
Chapter
9.
The Local-Global Approach to Lattices
175
§9.1.
Localization
176
§9.2.
The Genus
178
§9.3.
Maximal Lattices and the Cassels-Pfister Theorem
181
§9.4.
Sums of Integer Squares (Second Version)
184
Exercises
187
§9.5.
Indefinite Unimodular Z-Lattices
188
§9.6.
The Eichler-Kneser Theorem; the Lattice Zn
191
§9.7.
Growth of Class Numbers with Rank
196
§9.8.
Introduction to Neighbor Lattices
201
Exercises
205
Chapter
10.
Lattices over ¥q[x]
207
§10.1.
An Initial Example
209
§10.2.
Classification of Definite FjzJ-Lattices
210
§10.3.
On the Hasse-Minkowski Theorem over Fq(x)
218
§10.4.
Representation by Fjxj-Lattices
220
Exercises
223
Chapter
11.
Applications to Cryptography
225
§11.1.
A Brief Sketch of the Cryptographic Setting
225
§11.2.
Lattices in Rn
227
χ
Contents
§11.3. LLL-Reduction 230
§11.4.
Lattice Attacks on Knapsack Cryptosystems
235
§11.5.
Remarks on Lattice-Based Cryptosystems
239
Appendix: Further Reading
241
Bibliography
245
|
| adam_txt |
Contents
Preface
xi
Chapter
1.
A Brief Classical Introduction
1
§1.1.
Quadratic Forms as Polynomials
1
§1.2.
Representation and Equivalence; Matrix Connections;
Discriminants
4
Exercises
7
§1.3.
A Brief Historical Sketch, and Some References to the
Literature
7
Chapter
2.
Quadratic Spaces and Lattices
13
§2.1.
Fundamental Definitions
13
§2.2.
Orthogonal Splitting: Examples of Isometry and Non-isometry
16
Exercises
20
§2.3.
Representation. Splitting, and Isotropy; Invariants u(F) and
s(F)
21
§2.4.
The Orthogonal Group of a Space
26
§2.5.
Witt's Cancellation Theorem and Its Consequences
29
§2.6.
Witt's Chain Equivalence Theorem
34
§2.7.
Tensor Products of Quadratic Spaces; the Witt ring of a field
35
Exercises
39
§2.8.
Quadratic Spaces over Finite Fields
40
§2.9.
Hermitian Spaces
44
Exercises
49
vii
viii Contents
Chapter
3.
Valuations, Local Fields, and p-adic Numbers
51
§3.1.
Introduction to Valuations
51
§3.2.
Equivalence of Valuations; Prime Spots on a Field
54
Exercises
58
§3.3.
Completions, Qp, Residue Class Fields
59
§3.4.
Discrete Valuations
63
§3.5.
The Canonical Power Series Representation
64
§3.6.
Hensel's Lemma, the Local Square Theorem, and Local Fields
69
§3.7.
The Legendre Symbol; Recognizing Squares in Qp
74
Exercises
76
Chapter
4.
Quadratic Spaces over Qp
81
§4.1.
The Hubert Symbol
81
§4.2.
The
Hasse
Symbol (and an Alternative)
86
§4.3.
Classification of Quadratic Qp-Spaces
87
§4.4.
Hermitian Spaces over Quadratic Extensions of Qp
92
Exercises
94
Chapter
5.
Quadratic Spaces over
Q
97
§5.1.
The Product Formula and Hubert's Reciprocity Law
97
§5.2.
Extension of the Scalar Field
98
§5.3.
Local to Global: The Hasse-Minkowski Theorem
99
§5.4.
The Bruck-Ryser Theorem on Finite
Projective
Planes
105
§5.5.
Sums of Integer Squares (First Version)
109
Exercises 111
Chapter
6.
Lattices over Principal Ideal Domains
113
§6.1.
Lattice Basics
114
§6.2.
Valuations and Fractional Ideals
116
§6.3.
Invariant factors
118
§6.4.
Lattices on Quadratic Spaces
122
§6.5.
Orthogonal Splitting and Triple Diagonalization
124
§6.6.
The Dual of a Lattice
128
Exercises
130
§6.7.
Modular Lattices
133
§6.8.
Maximal Lattices
136
§6.9.
Unimodular Lattices and Pythagorean Triples
138
Contents ix
§6.10.
Remarks on Lattices over More General Rings
141
Exercises
142
Chapter
7.
Initial Integral Results
145
§7.1.
The Minimum of a Lattice; Definite Binary Z-Lattices
146
§7.2.
Hermite's Bound on
min L,
with a Supplement for fc[a;]-Lattices
149
§7.3.
Djokovic's Reduction of fcJxj-Lattices; Harder's Theorem
153
§7.4.
Finiteness of Class Numbers (The
Anisotropie
Case)
156
Exercises
158
Chapter
8.
Local Classification of Lattices
161
§8.1.
Jordan Splittings
161
§8.2.
Nondyadic Classification
164
§8.3.
Towards 2-adic Classification
165
Exercises
171
Chapter
9.
The Local-Global Approach to Lattices
175
§9.1.
Localization
176
§9.2.
The Genus
178
§9.3.
Maximal Lattices and the Cassels-Pfister Theorem
181
§9.4.
Sums of Integer Squares (Second Version)
184
Exercises
187
§9.5.
Indefinite Unimodular Z-Lattices
188
§9.6.
The Eichler-Kneser Theorem; the Lattice Zn
191
§9.7.
Growth of Class Numbers with Rank
196
§9.8.
Introduction to Neighbor Lattices
201
Exercises
205
Chapter
10.
Lattices over ¥q[x]
207
§10.1.
An Initial Example
209
§10.2.
Classification of Definite FjzJ-Lattices
210
§10.3.
On the Hasse-Minkowski Theorem over Fq(x)
218
§10.4.
Representation by Fjxj-Lattices
220
Exercises
223
Chapter
11.
Applications to Cryptography
225
§11.1.
A Brief Sketch of the Cryptographic Setting
225
§11.2.
Lattices in Rn
227
χ
Contents
§11.3. LLL-Reduction 230
§11.4.
Lattice Attacks on Knapsack Cryptosystems
235
§11.5.
Remarks on Lattice-Based Cryptosystems
239
Appendix: Further Reading
241
Bibliography
245 |
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| index_date | 2024-07-02T21:14:52Z |
| indexdate | 2024-07-09T21:17:12Z |
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| isbn | 9780821844656 0821844652 |
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| series2 | Graduate studies in mathematics |
| spelling | Gerstein, Larry J. 1940- Verfasser (DE-588)117710490 aut Basic quadratic forms Larry J. Gerstein Providence, R.I. American Mathematical Society 2008 XIII, 255 S. txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 90 Formes quadratiques Théorie des nombres Équations du second degré Forms, Quadratic Equations, Quadratic Number theory Quadratische Form (DE-588)4128297-8 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Quadratische Form (DE-588)4128297-8 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4704-1159-6 Graduate studies in mathematics 90 (DE-604)BV009739289 90 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016559985&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gerstein, Larry J. 1940- Basic quadratic forms Graduate studies in mathematics Formes quadratiques Théorie des nombres Équations du second degré Forms, Quadratic Equations, Quadratic Number theory Quadratische Form (DE-588)4128297-8 gnd |
| subject_GND | (DE-588)4128297-8 (DE-588)4123623-3 |
| title | Basic quadratic forms |
| title_auth | Basic quadratic forms |
| title_exact_search | Basic quadratic forms |
| title_exact_search_txtP | Basic quadratic forms |
| title_full | Basic quadratic forms Larry J. Gerstein |
| title_fullStr | Basic quadratic forms Larry J. Gerstein |
| title_full_unstemmed | Basic quadratic forms Larry J. Gerstein |
| title_short | Basic quadratic forms |
| title_sort | basic quadratic forms |
| topic | Formes quadratiques Théorie des nombres Équations du second degré Forms, Quadratic Equations, Quadratic Number theory Quadratische Form (DE-588)4128297-8 gnd |
| topic_facet | Formes quadratiques Théorie des nombres Équations du second degré Forms, Quadratic Equations, Quadratic Number theory Quadratische Form Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016559985&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV009739289 |
| work_keys_str_mv | AT gersteinlarryj basicquadraticforms |