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Gauss Sums. Jacobi Sums and Cyclotomic Numbers. Evaluation of Jacobi Sums Over Fp. Determination of Gauss Sums Over Fp. Difference Sets. Jacobsthal Sums Over Fp. Residuacity. Reciprocity Laws. Congruences for Binomial Coefficients. Diagonal Equations over Finite Fields. Gauss Sums over Fq. Eisenstein Sums. Brewer Sums. A General Eisenstein Reciprocity Law. Research Problems. Bibliography. Notation. Indexes.