Gauss Sums and Kloosterman Sums over Residue Rings of Algebraic Integers

Let O denote the ring of integers of an algebraic number field of degree m which is totally and tamely ramified at the prime p. Write ζq = exp(2πi/q), where q = pr. We evaluate the twisted Kloosterman sum ∑ α∈(O/qO)∗ χ(N(α))ζ T (α)+z/N(α) q , where T and N denote trace and norm, and where χ is a Dirichlet character (mod q). This extends results of Salié for m = 1 and of Yangbo Ye for prime m dividing p− 1. Our method is based upon our evaluation of the Gauss sum ∑ α∈(O/qO)∗ χ(N(α))ζ T (α) q , which extends results of Mauclaire for m = 1.