Quadratics:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Boca Raton u.a.
CRC Press
1996
|
| Schriftenreihe: | CRC Press series on discrete mathematics and its applications
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XX, 387 S. |
| ISBN: | 0849339839 |
Internformat
MARC
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| 245 | 1 | 0 | |a Quadratics |c Richard A. Mollin |
| 264 | 1 | |a Boca Raton u.a. |b CRC Press |c 1996 | |
| 300 | |a XX, 387 S. | ||
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| 490 | 0 | |a CRC Press series on discrete mathematics and its applications | |
| 650 | 4 | |a Corps quadratiques | |
| 650 | 7 | |a Corps quadratiques |2 ram | |
| 650 | 7 | |a Kwadratische systemen |2 gtt | |
| 650 | 4 | |a Quadratic fields | |
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Datensatz im Suchindex
| _version_ | 1804124980043055104 |
|---|---|
| adam_text | Contents
List of Symbols xi
Preface xv
Introduction xvii
1 Background from Algebraic Number Theory. 1
1.1 Quadratic Fields: Integers and Units 4
1.2 The Arithmetic of Ideals in Quadratic Fields 9
1.3 The Class Group and Class Number 14
1.4 Reduced Ideals 19
1.5 Quadratic Orders 23
1.6 Powerful Numbers: An Application of Real Quadratics 30
2 Continued Fractions Applied to Quadratic Fields. 41
2.1 Continued Fractions and Real Quadratics: The Infrastructure ... 41
2.2 The Continued Fraction Analogue for Complex Quadratics 63
3 Diophantine Equations and Class Numbers. 67
3.1 Class Numbers and Complex Quadratics 67
3.2 Real Quadratics and Diophantine Equations 77
3.3 Reduced Ideals and Diophantine Equations 87
3.4 Class Numbers and Real Quadratics 92
3.5 Halfway to a Solution 96
4 Prime Producing Polynomials. 105
4.1 Complex Prime Producers 105
4.2 Real Prime Producers 129
4.3 Density of Primes 145
5 Class Numbers: Criteria and Bounds. 149
5.1 Factoring Rabinowitsch 149
5.2 Class Number One Criteria 158
5.3 Class Number Bounds via the Divisor Function 163
5.4 The GRH: Relevance of the Riemann Hypothesis 172
6 Ambiguous Ideals. 187
6.1 Ambiguous Cycles in Real Orders: The Palindromic Index 187
6.2 Exponent Two 199
viii Contents
7 Influence of the Infrastructure. 223
7.1 Quadratic Residue Covers 223
7.2 Consecutive Powers 232
8 Algorithms. 235
8.1 Computation of the Class Number of a Real Quadratic Field .... 235
8.2 Cryptology 257
8.3 Implications of Computational Mathematics for the Philosophy of
Mathematics 266
APPENDIX A: Tables. 271
Table Al: This is a list of all positive fundamental radicands with class
number h — 1 and period length £, of the simple continued fraction
expansion of the principal class, less than 24, known to be uncondi¬
tionally complete whereas Table Al is complete with one GRH ruled
out exception, as are Tables A2 A4, A6 A7 and A9 271
Table A2: This is a subset of Table A1 with D= 1 (mod 8) 273
Table A3: hA = 2 for fundamental radicands D 0 with i 24 .... 274
Table A4: This is a list of all fundamental radicands of ERD type with
class groups of exponent 2, broken down into three parts depending
on congruence modulo 4 of the radicand 278
Table A5: This three part table is an illustration of a computer run
done for the proof of Theorem 6.2.2 281
Table A6: This is a list of all fundamental radicands D 0 of ERD type
having no split primes less than the Minkowski bound 283
Table A7: This is a complete list of all fundamental radicands D 0
with n^ ^ 0 (see Exercise 3.2.11) and associated regulators, such
that the class number is 1 284
Table A8: This is a list of all fundamental discriminants A = 1 (mod 8)
of ERD type with class number less than 24, and is known to be un¬
conditionally complete 285
Table A9: This table lists all fundamental discriminants of ERD type
with class number 2 286
APPENDIX B: Fundamental Units of Real Quadratic Fields.
This list is broken up into three parts according to congruence mod¬
ulo 4 of fundamental radicands less than 2 • 103 287
APPENDIX C: Class Numbers of Real Quadratic Fields.
This table is presented in matrix form with each entry describing a
specified class number together with the norm of the fundamental
unit with radicands less than 104 313
APPENDIX D: Class Numbers of Complex Quadratic Fields (and
their class group structure).
This is a table of fundamental radicands D 0 for |D| 2 ¦ 103,
together with h and the class group structure as a product of cyclic
groups 335
Contents ix
APPENDIX E: A Gazetteer of Forms
We give an overview of the classical theory of binary quadratic forms
and the interrelationships with ideal theory 347
APPENDIX F: Analytic Considerations
This provides background facts from analytic number theory. In
particular this gives a brief overview of the required material for
Chapter Five, section four 355
BIBLIOGRAPHY. 359
INDEX. 383
|
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| id | DE-604.BV010513610 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:54:17Z |
| institution | BVB |
| isbn | 0849339839 |
| language | English |
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| physical | XX, 387 S. |
| publishDate | 1996 |
| publishDateSearch | 1996 |
| publishDateSort | 1996 |
| publisher | CRC Press |
| record_format | marc |
| series2 | CRC Press series on discrete mathematics and its applications |
| spelling | Mollin, Richard A. 1947- Verfasser (DE-588)124930859 aut Quadratics Richard A. Mollin Boca Raton u.a. CRC Press 1996 XX, 387 S. txt rdacontent n rdamedia nc rdacarrier CRC Press series on discrete mathematics and its applications Corps quadratiques Corps quadratiques ram Kwadratische systemen gtt Quadratic fields HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007007484&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mollin, Richard A. 1947- Quadratics Corps quadratiques Corps quadratiques ram Kwadratische systemen gtt Quadratic fields |
| title | Quadratics |
| title_auth | Quadratics |
| title_exact_search | Quadratics |
| title_full | Quadratics Richard A. Mollin |
| title_fullStr | Quadratics Richard A. Mollin |
| title_full_unstemmed | Quadratics Richard A. Mollin |
| title_short | Quadratics |
| title_sort | quadratics |
| topic | Corps quadratiques Corps quadratiques ram Kwadratische systemen gtt Quadratic fields |
| topic_facet | Corps quadratiques Kwadratische systemen Quadratic fields |
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