Diophantine equations and power integral bases: new computational methods
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston, Mass. ; Basel ; Berlin
Birkhäuser
2002
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 171 - 180 |
| Physical Description: | XIII, 184 Seiten 24 cm |
| ISBN: | 3764342714 0817642714 |
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| 100 | 1 | |a Gaál, István |d 1960- |e Verfasser |0 (DE-588)123883768 |4 aut | |
| 245 | 1 | 0 | |a Diophantine equations and power integral bases |b new computational methods |c István Gaál |
| 264 | 1 | |a Boston, Mass. ; Basel ; Berlin |b Birkhäuser |c 2002 | |
| 300 | |a XIII, 184 Seiten |c 24 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Literaturverz. S. 171 - 180 | ||
| 650 | 4 | |a Algebraic fields | |
| 650 | 4 | |a Bases (Linear topological spaces) | |
| 650 | 4 | |a Diophantine equations | |
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| adam_text |
CONTENTS
PREFACE
XI
ACKNOWLEDGEMENTS
XV
1
INTRODUCTION
1
1.1
BASIC
CONCEPTS
.
1
1.2
RELATED
RESULTS
.
4
2
AUXILIARY
RESULTS,
TOOLS
7
2.1
BAKER
'
S
METHOD,
EFFECTIVE
FINITENESS
THEOREMS
.
8
2.2
REDUCTION
.
9
2.2.1
DAVENPORT
LEMMA
.
9
2.2.2
THE
GENERAL
CASE
.
10
2.3
ENUMERATION
METHODS
.
12
2.4
SOFTWARE,
HARDWARE
.
16
3
AUXILIARY
EQUATIONS
19
3.1
THUE
EQUATIONS
.
19
3.1.1
ELEMENTARY
ESTIMATES
.
20
3.1.2
THUE
'
S
THEOREM
.
20
3.1.3
FAST
ALGORITHM
FOR
FINDING
"SMALL"
SOLUTIONS
.
21
3.1.4
EFFECTIVE
METHODS
.
22
3.1.5
THE
METHOD
OF
BILU
AND
HANROT
.
23
3.2
INHOMOGENEOUS
THUE
EQUATIONS .
24
VIII
CONTENTS
3.2.1
ELEMENTARY
ESTIMATES
.
25
3.2.2
BAKER
'
S
METHOD
.
26
3.2.3
REDUCTION,
TEST
.
27
3.2.4
AN
ANALOGUE
OF
THE
BILU-HANROT
METHOD
.
27
3.3
RELATIVE
THUE
EQUATIONS
.
28
3.3.1
BAKER
'
S
METHOD,
REDUCTION
.
29
3.3.2
ENUMERATION
.
31
3.3.3
AN
EXAMPLE
.
32
3.4
THE
RESOLUTION
OF
NORM
FORM
EQUATIONS
.
34
3.4.1
PRELIMINARIES
.
34
3.4.2
SOLVING
THE
UNIT
EQUATION
.
36
3.4.3
CALCULATING
THE
SOLUTIONS
OF
THE
NORM
FORM
EQUATION
.
38
3.4.4
EXAMPLES
.
39
4
INDEX
FORM
EQUATIONS
IN
GENERAL
45
4.1
THE
STRUCTURE
OF
THE
INDEX
FORM
.
45
4.2
USING
RESOLVENTS
.
47
4.3
FACTORIZING
THE
INDEX
FORM
WHEN
PROPER
SUBFIELDS
EXIST
.
47
4.4
COMPOSITE
FIELDS
.
48
4.4.1
COPRIME
DISCRIMINANTS
.
48
4.4.2
NON-COPRIME
DISCRIMINANTS
.
50
5
CUBIC
FIELDS
53
5.1
ARBITRARY
CUBIC
FIELDS
.
53
5.2
SIMPLEST
CUBIC
FIELDS
.
54
6
QUARTIC
FIELDS
55
6.1
ALGORITHM
FOR
ARBITRARY
QUARTIC
FIELDS
.
56
6.1.1
THE
RESOLVENT
EQUATION
.
56
6.1.2
THE
QUARTIC
THUE
EQUATIONS
.
57
6.1.3
PROOF
OF
THE
THEOREM
ON
THE
QUARTIC
THUE
EQUATIONS
.
60
6.1.4
EXAMPLES
.
64
6.2 SIMPLEST
QUARTIC
FIELDS
.
65
6.3
AN
INTERESTING
APPLICATION
TO
MIXED
DIHEDRAL
QUARTIC
FIELDS
.
66
6.4
TOTALLY
COMPLEX
QUARTIC
FIELDS
.
67
6.4.1
PARAMETRIC
FAMILIES
OF
TOTALLY
COMPLEX
QUARTIC
FIELDS
.
68
6.5
BICYCLIC
BIQUADRATIC
NUMBER
FIELDS
.
70
6.5.1
INTEGRAL
BASIS,
INDEX
FORM
.
70
6.5.2
THE
TOTALLY
REAL
CASE
.
71
6.5.3
THE
TOTALLY
COMPLEX
CASE
.
74
6.5.4
THE
FIELD
INDEX
OF
BICYCLIC
BIQUADRATIC
NUMBER
FIELDS
.
75
7
QUINTIC
FIELDS
79
7.1
ALGORITHM
FOR
ARBITRARY
QUINTIC
FIELDS
.
79
7.1.1
PRELIMINARIES
.
80
CONTENTS
IX
7.1.2
BAKER
'
S
METHOD,
REDUCTION
.
81
7.1.3
ENUMERATION
.
82
7.1.4
EXAMPLES
.
84
7.2
LEHMER
'
S
QUINTICS
.
88
7.2.1
INTEGER
BASIS,
UNIT
GROUP
.
89
7.2.2
THE
INDEX
FORM
.
91
7.2.3
THE
INDEX
FORM
EQUATION
.
92
7.2.4
THE
EXCEPTIONAL
CASE
.
94
8
SEXTIC
FIELDS
97
8.1
SEXTIC
FIELDS
WITH
A
QUADRATIC
SUBFIELD
.
97
8.1.1
REAL
QUADRATIC
SUBFIELD
.
99
8.1.2
TOTALLY
REAL
SEXTIC
FIELDS
WITH
A
QUADRATIC
AND
A
CUBIC
SUBFIELD
.
100
8.1.3
IMAGINARY
QUADRATIC
SUBFIELD
.
101
8.1.4
SEXTIC
FIELDS
WITH
AN
IINAGINARY
QUADRATIC
AND
A
REAL
CUBIC
SUBFIELD
.
101
8.1.5
PARAMETRIC
FAMILIES
OF
SEXTIC
FIELDS
WITH
IMAGINARY
QUADRATIC
AND
REAL
CUBIC
SUBFIELDS
.
106
8.2
SEXTIC
FIELDS
WITH
A
CUBIC
SUBFIELD
.
109
8.3
SEXTIC
FIELDS
AS
COMPOSITE
FIELDS
.
110
8.3.1
A
CYCLIC
SEXTIC
FIELD
.
ILL
8.3.2
A
NON-CYCLIC
SEXTIC
FIELD
.
ILL
8.3.3
THE
PARAMETRIC
FAMILY
OF
SIMPLEST
SEXTIC
FIELDS
.
ILL
9
RELATIVE
POWER
INTEGRAL
BASES
113
9.1
BASIC
CONCEPTS
.
113
9.2
RELATIVE
CUBIC
EXTENSIONS
.
114
9.2.1
EXAMPLE
1.
CUBIC
EXTENSION
OF
A
QUINTIC
FIELD
.
115
9.2.2
EXAMPLE
2.
CUBIC
EXTENSION
OF
A
SEXTIC
FIELD
.
116
9.2.3
COMPUTATIONAL
EXPERIENCES
.
121
9.3
RELATIVE
QUARTIC
EXTENSIONS
.
121
9.3.1
PRELIMINARIES
.
121
9.3.2
THE
CUBIC
RELATIVE
THUE
EQUATION
.
121
9.3.3
REPRESENTING
THE
VARIABLES
AS
BINARY
QUADRATIC
FORMS
.
.
.
123
9.3.4
THE
QUARTIC
RELATIVE
THUE
EQUATIONS
.
124
9.3.5
AN
EXAMPLE
FOR
COMPUTING
RELATIVE
POWER
INTEGRAL
BASES
IN
A
FIELD
OF
DEGREE
12
WITH
A
CUBIC
SUBFIELD
.
125
10
SOME
HIGHER
DEGREE
FIELDS
129
10.1
OCTIC
FIELDS
WITH
A
QUADRATIC
SUBFIELD
.
129
10.1.1
PRELIMINARIES
.
129
10.1.2
THE
UNIT
EQUATION
.
131
10.1.3
THE
INHOMOGENEOUS
THUE
EQUATION
.
132
10.1.4
SIEVING
.
133
X
CONTENTS
10.1.5
AN
EXAMPLE
FOR
COMPUTING
POWER
INTEGRAL
BASES
IN
AN
OCTIC
FIELD
WITH
A
QUADRATIC
SUBFIELD
.
134
10.2
NONIC
FIELDS
WITH
CUBIC
SUBFIELDS
.
136
10.2.1
THE
RELATIVE
THUE
EQUATIONS
.
137
10.2.2
THE
UNIT
EQUATION
OVER
THE
NORMAL
CLOSURE
.
138
10.2.3
THE
COMMON
VARIABLES
.
140
10.2.4
EXAMPLES
.
143
10.3
SOME
MORE
FIELDS
OF
HIGHER
DEGREE
.
143
10.3.1
POWER
INTEGRAL
BASES
IN
IMAGINARY
QUADRATIC
EXTENSIONS
OF
TOTALLY
REAL
CYCLIC
FIELDS
OF
PRIME
DEGREE
.
144
10.3.2
POWER
INTEGRAL
BASES
IN
IMAGINARY
QUADRATIC
EXTENSIONS
OF
LEHMER
'
S
QUINTICS
.
146
10.3.3
ONE
MORE
COMPOSITE
FIELD
.
147
11
TABLES
149
11.1
CUBIC
FIELDS
.
149
11.1.1
TOTALLY
REAL
CUBIC
FIELDS
.
150
11.1.2
COMPLEX
CUBIC
FIELDS
.
152
11.2
QUARTIC
FIELDS
.
152
11.2.1
THE
DISTRIBUTION
OF
THE
MINIMAL
INDICES
.
153
11.2.2
THE
AVERAGE
BEHAVIOR
OF
THE
MINIMAL
INDICES
.
153
11.2.3
TOTALLY
REAL
CYCLIC
QUARTIC
FIELDS
.
154
11.2.4
MONOGENIC
MIXED
DIHEDRAL
EXTENSIONS
OF
REAL
QUADRATIC
FIELDS
.
155
11.2.5
TOTALLY
REAL
BICYCLIC
BIQUADRATIC
NUMBER
FIELDS
.
156
11.2.6
TOTALLY
COMPLEX
BICYCLIC
BIQUADRATIC
NUMBER
FIELDS
.
160
11.2.7
SOME
MORE
QUARTIC
FIELDS
.
161
11.3
SEXTIC
FIELDS
.
167
11.3.1
TOTALLY
REAL
CYCLIC
SEXTIC
FIELDS
.
167
11.3.2
SEXTIC
FIELDS
WITH
IMAGINARY
QUADRATIC
SUBFIELDS
.
168
REFERENCES
171
AUTHOR
INDEX
181
SUBJECT
INDEX
183 |
| any_adam_object | 1 |
| author | Gaál, István 1960- |
| author_GND | (DE-588)123883768 |
| author_facet | Gaál, István 1960- |
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| author_variant | i g ig |
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| callnumber-first | Q - Science |
| callnumber-label | QA242 |
| callnumber-raw | QA242 |
| callnumber-search | QA242 |
| callnumber-sort | QA 3242 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 180 SK 200 |
| ctrlnum | (OCoLC)49283460 (DE-599)BVBBV014570446 |
| dewey-full | 512/.72 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.72 |
| dewey-search | 512/.72 |
| dewey-sort | 3512 272 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV014570446 |
| illustrated | Not Illustrated |
| indexdate | 2024-08-22T00:30:14Z |
| institution | BVB |
| isbn | 3764342714 0817642714 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009906976 |
| oclc_num | 49283460 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-634 DE-188 |
| owner_facet | DE-355 DE-BY-UBR DE-634 DE-188 |
| physical | XIII, 184 Seiten 24 cm |
| publishDate | 2002 |
| publishDateSearch | 2002 |
| publishDateSort | 2002 |
| publisher | Birkhäuser |
| record_format | marc |
| spelling | Gaál, István 1960- Verfasser (DE-588)123883768 aut Diophantine equations and power integral bases new computational methods István Gaál Boston, Mass. ; Basel ; Berlin Birkhäuser 2002 XIII, 184 Seiten 24 cm txt rdacontent n rdamedia nc rdacarrier Literaturverz. S. 171 - 180 Algebraic fields Bases (Linear topological spaces) Diophantine equations Algebraischer Körper (DE-588)4141852-9 gnd rswk-swf Diophantische Gleichung (DE-588)4012386-8 gnd rswk-swf Basis Mathematik (DE-588)4228195-7 gnd rswk-swf Diophantische Gleichung (DE-588)4012386-8 s DE-604 Algebraischer Körper (DE-588)4141852-9 s Basis Mathematik (DE-588)4228195-7 s DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009906976&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gaál, István 1960- Diophantine equations and power integral bases new computational methods Algebraic fields Bases (Linear topological spaces) Diophantine equations Algebraischer Körper (DE-588)4141852-9 gnd Diophantische Gleichung (DE-588)4012386-8 gnd Basis Mathematik (DE-588)4228195-7 gnd |
| subject_GND | (DE-588)4141852-9 (DE-588)4012386-8 (DE-588)4228195-7 |
| title | Diophantine equations and power integral bases new computational methods |
| title_auth | Diophantine equations and power integral bases new computational methods |
| title_exact_search | Diophantine equations and power integral bases new computational methods |
| title_full | Diophantine equations and power integral bases new computational methods István Gaál |
| title_fullStr | Diophantine equations and power integral bases new computational methods István Gaál |
| title_full_unstemmed | Diophantine equations and power integral bases new computational methods István Gaál |
| title_short | Diophantine equations and power integral bases |
| title_sort | diophantine equations and power integral bases new computational methods |
| title_sub | new computational methods |
| topic | Algebraic fields Bases (Linear topological spaces) Diophantine equations Algebraischer Körper (DE-588)4141852-9 gnd Diophantische Gleichung (DE-588)4012386-8 gnd Basis Mathematik (DE-588)4228195-7 gnd |
| topic_facet | Algebraic fields Bases (Linear topological spaces) Diophantine equations Algebraischer Körper Diophantische Gleichung Basis Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009906976&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT gaalistvan diophantineequationsandpowerintegralbasesnewcomputationalmethods |