Gauss and Jacobi Sums,9780471128076

Gauss and Jacobi Sums

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Edition: 1st
ISBN13: 9780471128076
ISBN10: 0471128074
Format: Hardcover
Pub. Date: 1998-06-01
Publisher(s): Wiley-Interscience
List Price: $245.27

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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

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