Gauss and Jacobi Sums

Devised in the 19th century, Gauss and Jacobi Sums are classical formulas that form the basis for contemporary research in many of today's sciences. This book offers readers a solid grounding on the origin of these abstract, general theories. Though the main focus is on Gauss and Jacobi, the book does explore other relevant formulas, including Cauchy.

Bruce Carl Berndt is an American mathematician. Berndt attended college at Albion College, graduating in 1961, where he also ran track. He received his master's and doctoral degrees from the University of Wisconsin-Madison. Ronald J. Evans is the author of Gauss and Jacobi Sums, published by Wiley.

Introduction

1

(6)

1 Gauss Sums

7

(50)

1.1 Elementary Properties of Gauss sums over F(q)

7

(5)

1.2 The reciprocity theorem for quadratic Gauss sums

12

(6)

1.3 Gauss' evaluation of a quadratic Gauss sum

18

(6)

1.4 Estermann's evaluation of a quadratic Gauss sum

24

(1)

1.5 Elementary determination of quadratic Gauss sums

25

(3)

1.6 Gauss character sums over the ring of integers (mod k)

28

(14)

Exercises 1

42

(8)

Notes on Chapter 1

50

(7)

2 Jacobi Sums and Cyclotomic Numbers

57

(43)

2.1 Basic properties of Jacobi sums over F(q)

57

(11)

2.2 Cyclotomic numbers

68

(3)

2.3 Cyclotomic numbers of order 3

71

(3)

2.4 Cyclotomic numbers of order 4

74

(5)

2.5 Relationship between Jacobi sums and cyclotomic numbers

79

(2)

2.6 Determination of ind(g)2 and ind(g)k (mod k)

81

(6)

2.7 Generalized cyclotomic numbers and the determination of ind(g)l (mod k)

87

(5)

Exercises 2

92

(5)

Notes on Chapter 2

97

(3)

3 Evaluation of Jacobi Sums over F(p)

100

(53)

3.1 Cubic and sextic sums

103

(4)

3.2 Quartic sums

107

(2)

3.3 Octic sums

109

(2)

3.4 Bioctic sums

111

(4)

3.5 Duodecic sums

115

(3)

3.6 Biduodecic sums

118

(6)

3.7 Quintic and decic sums

124

(12)

3.8 Bidecic sums

136

(4)

3.9 Septic sums

140

(7)

Exercises 3

147

(3)

Notes on Chapter 3

150

(3)

4 Determination of Gauss Sums over F(p)

153

(21)

4.1 The Gauss sums g(3) and g(6)

154

(6)

4.2 The Gauss sum g(4)

160

(4)

4.3 The Gauss sum g(8)

164

(2)

4.4 The Gauss sum g(12)

166

(2)

Exercises 4

168

(3)

Notes on Chapter 4

171

(3)

5 Difference Sets

174

(9)

5.1 Basic definitions

174

(1)

5.2 Necessary and sufficient conditions for power residue difference sets

175

(1)

5.3 Applications of Gauss sums

176

(4)

Exercises 5

180

(1)

Notes on Chapter 5

181

(2)

6 Jacobsthal Sums over F(p)

183

(29)

6.1 Jacobsthal sums and their elementary properties

184

(5)

6.2 Explicit determination of some Jacobsthal sums

189

(7)

6.3 Applications to the distribution of quadratic residues and nonresidues

196

(3)

6.4 Congruences for Jacobsthal sums

199

(3)

6.5 Double Jacobsthal sums

202

(3)

Exercises 6

205

(4)

Notes on Chapter 6

209

(3)

7 Residuacity

212

(22)

7.1 Cubic residuacity

212

(4)

7.2 Quartic residuacity

216

(2)

7.3 Octic residuacity

218

(3)

7.4 Quintic residuacity

221

(4)

7.5 The quartic, octic, and bioctic character of 2

225

(5)

Exercises 7

230

(1)

Notes on Chapter 7

231

(3)

8 Reciprocity Laws

234

(34)

8.1 Cubic reciprocity

234

(7)

8.2 Biquadratic reciprocity

241

(10)

8.3 Rational reciprocity laws

251

(10)

Exercises 8

261

(3)

Notes on Chapter 8

264

(4)

9 Congruences for Binomial Coefficients

268

(26)

9.1 Binomial coefficients and Jacobi sums

268

(1)

9.2 Binomial coefficients modulo p

269

(7)

9.3 Binomial coefficients, Jacobi sums, and p-adic gamma functions

276

(4)

9.4 Binomial coefficients modulo p^2

280

(8)

Exercises 9

288

(3)

Notes on Chapter 9

291

(3)

10 Diagonal Equations over Finite Fields

294

(48)

10.1 Generalized Jacobi sums

295

(3)

10.2 A reduction formula for generalized Jacobi sums

298

(3)

10.3 Generalized Jacobi sums and Gauss sums

301

(2)

10.4 Number of solutions of the equation a(1)x(1)^(k)1+...+ a(n)x^(k)(n)(n) = a

303

(2)

10.5 Number of solutions of the equation a(1)x^2(1) +...+ a(n)x^2(n) = a

305

(2)

10.6 Number of solutions of the congruence A(1)x^3(1) +...+ A(n)x^3(n) = A (mod p)

307

(7)

10.7 Number of solutions of the congruence A(1)x^4(1) +...+ A(n)x^4(n) = A (mod p)

314

(4)

10.8 Bounds for the number of solutions

318

(4)

10.9 Generalized cyclotomic numbers and the congruence A(1)x^k(1) +...+ A(n)x^k(n) = A (mod p)

322

(4)

10.10 f-nominal periods and the period polynomial

326

(7)

Exercises 10

333

(3)

Notes on Chapter 10

336

(6)

11 Gauss Sums over F(q)

342

(47)

11.1 Prime ideal factorization of p

342

(2)

11.2 Stickelberger's congruence for Gauss sums

344

(7)

11.3 The Davenport-Hasse product formula

351

(4)

11.4 Restrictions and lifts of characters

355

(3)

11.5 The Davenport-Hasse theorem on lifted Gauss sums

358

(4)

11.6 Pure Gauss sums

362

(6)

11.7 Irreducible cyclic codes

368

(13)

Exercises 11

381

(3)

Notes on Chapter 11

384

(5)

12 Eisenstein Sums

389

(51)

12.1 Properties of the Eisenstein sum E(r)(X)

391

(5)

12.2 An Eisenstein sum over F(p)^2

396

(4)

12.3 Eisenstein sums of order 3

400

(2)

12.4 Eisenstein sums of order 4

402

(1)

12.5 Eisenstein sums of order 5

403

(2)

12.6 Eisenstein sums of order 6

405

(2)

12.7 Eisenstein sums of order 8

407

(5)

12.8 Some Eisenstein sums of order 7

412

(2)

12.9 Congruences of Eisenstein for binomial coefficients

414

(7)

12.10 Gauss sums and f-nomial periods over F(q)

421

(7)

12.11 An Eisenstein sum of order 20

428

(2)

12.12 An Eisenstein sum of order 16

430

(3)

12.13 An Eisenstein sum of order 12

433

(1)

Exercises 12

434

(4)

Notes on Chapter 12

438

(2)

13 Brewer Sums

440

(28)

13.1 Dickson polynomials

440

(3)

13.2 Formulas for the ordinary Brewer sums (XXX)

443

(7)

13.3 Evaluation of the Brewer sums (XXX)1, (XXX)2, (XXX)3, (XXX)4, and (XXX)6

450

(4)

13.4 Evaluation of the Brewer sums (XXX)5 and (XXX)10

454

(3)

13.5 Evaluation of the Brewer sum (XXX)8

457

(1)

13.6 Formulas for the generalized Brewer sums A(n)(a)

458

(2)

13.7 Evaluation of the Brewer sums (XXX)1(a), (XXX)2(a), (XXX)3(a), (XXX)4(a), and (XXX)6(a)

460

(4)

Exercises 13

464

(2)

Notes on Chapter 13

466

(2)

14 A General Eisenstein Reciprocity Law

468

(28)

14.1 A quotient of Gauss sums

468

(2)

14.2 Primary integers of cyclotomic fields

470

(4)

14.3 Statement of Eisenstein's reciprocity law

474

(1)

14.4 A general reciprocity relation

474

(3)

14.5 Proof of Eisenstein's reciprocity law for l > 2

477

(6)

14.6 Proof of Eisenstein's reciprocity law for l = 2

483

(7)

14.7 Application of Eisenstein's reciprocity law to Wieferich's theorem

490

(2)

Exercises 14

492

(2)

Notes on Chapter 14

494

(2)

Research Problems

496

(3)

Bibliography

499

(66)

Notation

565

(6)

Author Index

571

(6)

Subject Index

577

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Gauss and Jacobi Sums

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Author Biography

Table of Contents

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