Gauss and Jacobi sums:
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Wiley
1998
|
| Schriftenreihe: | Canadian Mathematical Society: Canadian Mathematical Society series of monographs and advanced texts
21 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XI, 583 S. |
| ISBN: | 0471128074 |
Internformat
MARC
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| 245 | 1 | 0 | |a Gauss and Jacobi sums |c Bruce C. Berndt ; Ronald J. Evans ; Kenneth S. Williams |
| 264 | 1 | |a New York [u.a.] |b Wiley |c 1998 | |
| 300 | |a XI, 583 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Canadian Mathematical Society: Canadian Mathematical Society series of monographs and advanced texts |v 21 | |
| 490 | 0 | |a A Wiley-Interscience publication | |
| 650 | 4 | |a Gauss, Sommes de | |
| 650 | 7 | |a Gauss-sommen |2 gtt | |
| 650 | 4 | |a Jacobi, Sommes de | |
| 650 | 4 | |a Gaussian sums | |
| 650 | 4 | |a Jacobi sums | |
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| 700 | 1 | |a Williams, Kenneth S. |d 1940- |e Verfasser |0 (DE-588)132012030 |4 aut | |
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Datensatz im Suchindex
| _version_ | 1804126503305216000 |
|---|---|
| adam_text | Contents
Introduction 1
1 Gauss Sums 7
1.1 Elementary properties of Gauss sums over Fq, 7
1.2 The reciprocity theorem for quadratic Gauss sums, 12
1.3 Gauss evaluation of a quadratic Gauss sum, 18
1.4 Estermann s evaluation of a quadratic Gauss sum, 24
1.5 Elementary determination of quadratic Gauss sums, 25
1.6 Gauss character sums over the ring of integers (mod k), 28
Exercises 1, 42
Notes on Chapter 1, 50
2 Jacobi Sums and Cyclotomic Numbers 57
2.1 Basic properties of Jacobi sums over Fq, 57
2.2 Cyclotomic numbers, 68
2.3 Cyclotomic numbers of order 3, 71
2.4 Cyclotomic numbers of order 4, 74
2.5 Relationship between Jacobi sums and cyclotomic numbers, 79
2.6 Determination of indA,2 and ind^ (mod k), 81
2.7 Generalized cyclotomic numbers and the determination of
incy(mod£), 87
Exercises 2, 92
Notes on Chapter 2, 97
3 Evaluation of Jacobi Sums over Fp 100
3.1 Cubic and sextic sums, 103
3.2 Quartic sums, 107
3.3 Octic sums, 109
3.4 Bioctic sums, 111
3.5 Duodecic sums, 115
3.6 Biduodecic sums, 118
viii Contents
3.7 Quintic and deck sums, 124
3.8 Bidecic sums, 136
3.9 Septic sums, 140
Exercises 3, 147
Notes on Chapter 3, 150
4 Determination of Gauss Sums over Fp 153
4.1 The Gauss sums #(3) and g{6), 154
4.2 The Gauss sum #(4), 160
4.3 The Gauss sum #(8), 164
4.4 The Gauss sum ^(12), 166
Exercises 4, 168
Notes on Chapter 4, 171
5 Difference Sets 174
5.1 Basic definitions, 174
5.2 Necessary and sufficient conditions for power residue
difference sets, 175
5.3 Applications of Gauss sums, 176
Exercises 5, 180
Notes on Chapter 5, 181
6 Jacobsthal Sums over Fp 183
6.1 Jacobsthal sums and their elementary properties, 184
6.2 Explicit determination of some Jacobsthal sums, 189
6.3 Applications to the distribution of quadratic residues
and nonresidues, 196
6.4 Congruences for Jacobsthal sums, 199
6.5 Double Jacobsthal sums, 202
Exercises 6, 205
Notes on Chapter 6, 209
7 Residuacity 212
7.1 Cubic residuacity, 212
7.2 Quartic residuacity, 216
7.3 Octic residuacity, 218
Contents ix
7.4 Quintic residuacity, 221
7.5 The quartic, octic, and bioctic character of 2, 225
Exercises 7, 230
Notes on Chapter 7, 231
8 Reciprocity Laws 234
8.1 Cubic reciprocity, 234
8.2 Biquadratic reciprocity, 241
8.3 Rational reciprocity laws, 251
Exercises 8, 261
Notes on Chapter 8, 264
9 Congruences for Binomial Coefficients 268
9.1 Binomial coefficients and Jacobi sums, 268
9.2 Binomial coefficients modulo p, 269
9.3 Binomial coefficients, Jacobi sums, and / adic gamma functions, 276
9.4 Binomial coefficients modulo p2, 280
Exercises 9, 288
Notes on Chapter 9, 291
10 Diagonal Equations over Finite Fields 294
10.1 Generalized Jacobi sums, 295
10.2 A reduction formula for generalized Jacobi sums, 298
10.3 Generalized Jacobi sums and Gauss sums, 301
10.4 Number of solutions of the equation aAx l + f a,,x* = a, 303
10.5 Number of solutions of the equation axx] + ¦ ¦ ¦ + anx;, a, 305
10.6 Number of solutions of the congruence Axx^ + Anxl =
A (mod/)), 307
10.7 Number of solutions of the congruence Ax x h v A,,xf, =
A (mod/)), 314
10.8 Bounds for the number of solutions, 318
10.9 Generalized cyclotomic numbers and the congruence
A^i + +A^f^Aimodp), 322
10.10 / nomial periods and the period polynomial, 326
Exercises 10, 333
Notes on Chapter 10, 336
x Contents
11 Gauss Sums over Fq 342
11.1 Prime ideal factorization of p, 342
11.2 Stickelberger s congruence for Gauss sums, 344
11.3 The Davenport Hasse product formula, 351
11.4 Restrictions and lifts of characters, 355
11.5 The Davenport Hasse theorem on lifted Gauss sums, 358
11.6 Pure Gauss sums, 362
11.7 Irreducible cyclic codes, 368
Exercises 11, 381
Notes on Chapter 11, 384
12 Eisenstein Sums 389
12.1 Properties of the Eisenstein sum Er(x), 391
12.2 An Eisenstein sum over Fp2, 396
12.3 Eisenstein sums of order 3, 400
12.4 Eisenstein sums of order 4, 402
12.5 Eisenstein sums of order 5, 403
12.6 Eisenstein sums of order 6, 405
12.7 Eisenstein sums of order 8, 407
12.8 Some Eisenstein sums of order 7, 412
12.9 Congruences of Eisenstein for binomial coefficients, 414
12.10 Gauss sums and/ nomial periods over Fq, 421
12.11 An Eisenstein sum of order 20, 428
12.12 An Eisenstein sum of order 16, 430
12.13 An Eisenstein sum of order 12, 433
Exercises 12, 434
Notes on Chapter 12, 438
13 Brewer Sums 440
13.1 Dickson polynomials, 440
13.2 Formulas for the ordinary Brewer sums A,,, 443
13.3 Evaluation of the Brewer sums A1( A2, A3, A4, and A6, 450
13.4 Evaluation of the Brewer sums A5 and A10, 454
13.5 Evaluation of the Brewer sum A8, 457
13.6 Formulas for the generalized Brewer sums An(a), 458
13.7 Evaluation of the Brewer sums A^a), A2(fl), A3(a), A4(a),
and A6(a), 460
Contents xi
Exercises 13, 464
Notes on Chapter 13, 466
14 A General Eisenstein Reciprocity Law 468
14.1 A quotient of Gauss sums, 468
14.2 Primary integers of cyclotomic fields, 470
14.3 Statement of Eisenstein s reciprocity law, 474
14.4 A general reciprocity relation, 474
14.5 Proof of Eisenstein s reciprocity law for / 2, 477
14.6 Proof of Eisenstein s reciprocity law for 1 2, 483
14.7 Application of Eisenstein s reciprocity law to Wieferich s theorem, 490
Exercises 14, 492
Notes on Chapter 14, 494
Research Problems 496
Bibliography 499
Notation 565
Author Index 571
Subject Index 577
|
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| series | Canadian Mathematical Society: Canadian Mathematical Society series of monographs and advanced texts |
| series2 | Canadian Mathematical Society: Canadian Mathematical Society series of monographs and advanced texts A Wiley-Interscience publication |
| spelling | Berndt, Bruce C. 1939- Verfasser (DE-588)133753441 aut Gauss and Jacobi sums Bruce C. Berndt ; Ronald J. Evans ; Kenneth S. Williams New York [u.a.] Wiley 1998 XI, 583 S. txt rdacontent n rdamedia nc rdacarrier Canadian Mathematical Society: Canadian Mathematical Society series of monographs and advanced texts 21 A Wiley-Interscience publication Gauss, Sommes de Gauss-sommen gtt Jacobi, Sommes de Gaussian sums Jacobi sums Gauß-Summe (DE-588)4156109-0 gnd rswk-swf Gauß-Summe (DE-588)4156109-0 s DE-604 Evans, Ronald J. Verfasser aut Williams, Kenneth S. 1940- Verfasser (DE-588)132012030 aut Canadian Mathematical Society: Canadian Mathematical Society series of monographs and advanced texts 21 (DE-604)BV012067942 21 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008053653&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Berndt, Bruce C. 1939- Evans, Ronald J. Williams, Kenneth S. 1940- Gauss and Jacobi sums Canadian Mathematical Society: Canadian Mathematical Society series of monographs and advanced texts Gauss, Sommes de Gauss-sommen gtt Jacobi, Sommes de Gaussian sums Jacobi sums Gauß-Summe (DE-588)4156109-0 gnd |
| subject_GND | (DE-588)4156109-0 |
| title | Gauss and Jacobi sums |
| title_auth | Gauss and Jacobi sums |
| title_exact_search | Gauss and Jacobi sums |
| title_full | Gauss and Jacobi sums Bruce C. Berndt ; Ronald J. Evans ; Kenneth S. Williams |
| title_fullStr | Gauss and Jacobi sums Bruce C. Berndt ; Ronald J. Evans ; Kenneth S. Williams |
| title_full_unstemmed | Gauss and Jacobi sums Bruce C. Berndt ; Ronald J. Evans ; Kenneth S. Williams |
| title_short | Gauss and Jacobi sums |
| title_sort | gauss and jacobi sums |
| topic | Gauss, Sommes de Gauss-sommen gtt Jacobi, Sommes de Gaussian sums Jacobi sums Gauß-Summe (DE-588)4156109-0 gnd |
| topic_facet | Gauss, Sommes de Gauss-sommen Jacobi, Sommes de Gaussian sums Jacobi sums Gauß-Summe |
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