本书主要包括高斯和的定义，雅可比和与分圆数，在F之上的雅可比估计，差集，偏差，互易律，二项式系数的同余，有限域对角方程，三、四、五、六、八阶的爱因斯坦和，斯蒂克伯格对高斯和的全等性的解释，剩余功率的充要条件，高斯和的应用，雅各布-斯涛尔和的同余，四次互逆律等内容。本书的目标是为读者提供关于高斯和与雅可比和的奇妙定理的简单应用，使读者易于使用，特别是对初级研究生来说，希望高斯，雅可比，艾森斯坦和其他数学家的经典定理能够更容易被理解。

Introduction
1 Gauss Sums
1.1 Elementary properties of Gauss sums over Fq,
1.2 The reciprocity theorem for quadratic Gauss sums,
1.3 Gauss' evaluation of a quadratic Gauss sum,
1.4 Estermann's evaluation of a quadratic Gauss sum,
1.5 Elementary determination of quadratic Gauss sums,
1.6 Gauss character sums over the ring of integers (rood k),
Exercises 1,
Notes on Chapter 1,
2 Jacobi Sums and Cyclotomic Numbers
2.1 Basic properties of Jacobi sums over F~,
2.2 Cyclotomic numbers,
2.3 Cyclotomic numbers of order 3,
2.4 Cyclotomic numbers of order 4,
2.5 Relationship between Jacobi sums and cyclotomic numbers,
2.6 Determination of indg2 and indgk (mod k),
2.7 Generalized cyclotomic numbers and the determination of ind,/(mod k),
Exercises 2,
Notes on Chapter 2,
3 Evaluation of Jacobi Sums over Fp
3.1 Cubic and sextic sums,
3.2 Quartic sums,
3.3 Octic sums,
3.4 Bioctic sums,
3.5 Duodecic sums,
3.6 Biduodecic sums,
3.7 Quintic and decic sums,
3.8 Bidecic sums,
3.9 Septic sums,
Exercises 3,
Notes on Chapter 3,
4 Determination of Gauss Sums over Fp
4.1 The Gauss sumsg(3) andg(6),
4.2 The Gauss sum g(4),
4.3 The Gauss sum g(8),
4.4 The Gauss sum g(12),
Exercises 4,
Notes on Chapter 4,
5 Difference Sets
5.1 Basic definitions,
5.2 Necessary and sufficient conditions for power residue
difference sets,
5.3 Applications of Gauss sums,
Exercises 5,
Notes on Chapter 5,
6 Jacobsthal Sums over Fp
6.1 Jacobsthal sums and their elementary properties,
6.2 Explicit determination of some Jacobsthal sums,
6.3 Applications to the distribution of quadratic residues
and nonresidues,
6.4 Congruences for Jacobsthal sums,
6.5 Double Jacobsthal sums,
Exercises 6,
Notes on Chapter 6,
7 Residuacity
7.1 Cubic residuacity,
7.2 Quartic residuacity,
7.3 Octic residuacity,
7.4 Quintic residuacity,
7.5 The quartic, octic, and bioctic character of 2,
Exercises 7,
Notes on Chapter 7,
8 Reciprocity Laws
8.1 Cubic reciprocity,
8.2 Biquadratic reciprocity,
8.3 Rational reciprocity laws,
Exercises 8,
Notes on Chapter 8,
9 Congruences for Binomial Coefficients
9.1 Binomial coefficients and Jacobi sums,
9.2 Binomial coefficients modulo p,
9.3 Binomial coefficients, Jacobi sums, and p-adic gamma functions,
9.4 Binomial coefficients modulo p2,
Exercises 9,
Notes on Chapter 9,
10 Diagonal Equations over Finite Fields
10.1 Generalized Jacobi sums,
10.2 A reduction formula for generalized Jacobi sums,
10.3 Generalized Jacobi sums and Gauss sums
10.4 Number of solutions of the equation α1χ1κ1+……αnχnκn
10.5 Number of solutions of the equation α1χ21+……αnχknn=α
10.6 Number of solutions of the congruence A 1 χ31+……+AnX3N= A (modp),
10.7 Number of solutions of the congruence A 1 χ41+……+AnX4N= A (modp),
10.8 Bounds for the number of solutions,
10.9 Generalized cyclotomic numbers and the congruenceA 1 χk1+……+AnXkN= A (modp),
10.10 f-nomial periods and the period polynomial,
Exercises 10,
Notes on Chapter 10,
11 Gauss Sums over Fq
11.1 Prime ideal factorization of p,
11.2 Stickelberger's congruence for Gauss sums,
11.3 The Davenport-Hasse product formula,
11.4 Restrictions and lifts of characters,
11.5 The Davenport-Hasse theorem on lifted Gauss sums,
11.6 Pure Gauss sums,
11.7 Irreducible cyclic codes,
Exercises 11,
Notes on Chapter 11,
12 Eisenstein Sums
12.1 Properties of the Eisenstein sum E,(X),
12.2 An Eisenstein sum over Fp2,
12.3 Eisenstein sums of order 3,
12.4 Eisenstein sums of order 4,
12.5 Eisenstein sums of order 5,
12.6 Eisenstein sums of order 6,
12.7 Eisenstein sums of order 8,
12.8 Some Eisenstein sums of order 7,
12.9 Congruences of Eisenstein for binomial coefficients,
12.10