Gauss and Jacobi Sums

by Berndt, Bruce C.; Evans, Ronald J.; Williams, Kenneth S.

• ISBN13: 9780471128076
• ISBN10: 0471128074
• Edition: 1st
• Format: Hardcover
• Copyright: 1998-06-01
• Publisher: Wiley-Interscience

Summary

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.

Author Biography

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.

Table of Contents

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