Berndt B.C., Evans R.J., Williams K.S. Gauss and Jacobi Sums

Издательство John Wiley, 1998, -598 pp.The origins of this book lie in the work of С.F. Gauss and C.G.J. Jacobi. Gauss introduced the Gauss sum in his Disquisitiones Arithmeticae in July, 1801, and Jacobi introduced the Jacobi sum in a letter to Gauss dated February 8, 1827.
The sum introduced by Gauss in 1801 is now called a (quadratic) Gauss sum. This sum is not easy to evaluate, even in the special case that m=1 and k is an odd positive integer. Specific examples convinced Gauss that the plus sign is always correct. On August 30, 1805, Gauss wrote in his diary that he devoted some time to this problem every week for more than four years before he was able to prove his conjecture on the sign of these sums. A few years later, Gauss [3] published an evaluation of his quadratic Gauss sum for all positive integers k. The sum now called a Jacobi sum, which is in essence the one Jacobi introduced in 1827.
The main prerequisites for the book are knowledge of undergraduate (including finite fields) and basic material in elementary and algebraic number theory. The reader who wishes to review the theory of finite fields may consult the texts of Lidl and Niederreiter or Ireland and Rosen. A small amount of complex analysis is required in Section 1.2, and some p-adic analysis is used in Sections 1.6, 9.3, and 11.2.Gauss Sums
Jacobi Sums and Cyclotomic Numbers
Evaluation of Jacobi Sums over F_p
Determination of Gauss Sums over F_p
Difference Sets
Jacobsthal Sums over F_p
Residuacity
Reciprocity Laws
Congruences for Binomial Coefficients
Diagonal Equations over Finite Fields
Gauss Sums over F_q
Eisenstein Sums
Brewer Sums
A General Eisenstein Reciprocity Law