Gauss Sums, Jacobi Sums, and Relative Gauss Sums

This chapter focuses on Gauss sums, Jacobi sums, cyclotomic numbers, relative Gauss sums, and prime ideal factorization of Gauss sums. Gauss sums may be determined explicitly for special characters. The chapter presents the congruence relations which are helpful for the evaluation of Jacobi sums. The concept of the relative Gauss sum was first introduced in the paper by Yamamoto and Yamada to construct a Williamson Hadamard matrix. Momihara constructed a family of skew Hadamard difference sets from normalized relative Gauss sums. The chapter gives a prime ideal factorization of a prime number p in a cyclotomic field Q(ζ_m).