GAUSS  AND  JACOBI   SUMS

by B.C. Berndt,  R.J. Evans,  and  K.S. Williams

Wiley-Interscience, N.Y., 1998
 



Table of Contents

Introduction    1

1    Gauss Sums
1.1    Elementary properties of Gauss sums over Fq    7
1.2    The reciprocity theorem for quadratic Gauss sums    12
1.3    Gauss's  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
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  indg 2 and  indg (mod k)    81
2.7    Generalized cyclotomic numbers and the determination of indg (mod k)    87
         Exercises 2    92
         Notes on Chapter 2    97

3    Evaluation of Jacobi Sums over Fp
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
3.7    Quintic and decic 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
4.1    The Gauss sums  g(3)  and g(6)    154
4.2    The Gauss sum   g(4)    160
4.3    The Gauss sum   g(8)    164
4.4    The Gauss sum   g(12)    166
         Exercises 4    168
         Notes on Chapter 4    171

5    Difference Sets
5.1    Basic definition    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
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
7.1    Cubic residuacity    212
7.2    Quartic residuacity    216
7.3    Octic residuacity    218
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
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
9.1    Binomial coefficients and Jacobi sums    268
9.2    Binomial coefficients modulo  p    269
9.3    Binomial coefficients, Jacobi sums, and  p-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
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
             a1 x1 k1   + . . . +  an xn kn  a    303
10.5    Number of solutions of the equation
             a1 x12     + . . . an xn2  a    305
10.6    Number of solutions of the congruence
            A1 x13    + . . . +   An xn3  A   (mod  p)    307
10.7    Number of solutions of the congruence
            A1 x14     + . . . +   An xn4  A   (mod  p)    314
10.8    Bounds for the number of solutions    318
10.9    Generalized cyclotomic numbers and the congruence
            A1 x1k    + . . . +   An xnk  =  A   (mod  p)    322
10.10   f-nomial periods and the period polynomial    326
           Exercises 10    333
           Notes on Chapter 10    336

11    Gauss Sums over  Fq
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
12.1      Properties of the Eisenstein sum  Er    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  f-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
13.1   Dickson polynomials    440
13.2   Formulas for the ordinary Brewer sums of order  n    443
13.3   Evaluation of the Brewer sums of orders 1, 2, 3, 4, and 6    450
13.4   Evaluation of the Brewer sums of orders 5 and 10    454
13.5   Evaluation of the Brewer sum of order 8    457
13.6   Formulas for the generalized Brewer sums of order  n    458
13.7   Evaluation of the generalized Brewer sums of orders 1, 2, 3, 4, and 6    460
          Exercises 13    464
          Notes on Chapter 13    466

14   A General Eisenstein Reciprocity Law
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  l 2    477
14.6   Proof of Eisenstein's reciprocity law for  l = 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

Errata, Remarks, and Reviews

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