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For any integer m > 1 x m = exp(2 i=m), and let Zm denote the group of reduced residues modulo m. Let q = p , a power of a prime p. The hyper- Kloosterman sums of dimension n > 0 are dened for q by R(d; q) = X x1;:::;xn2Z q x1+ +xn+d(x1 xn) 1 q (d 2 Z q); where x 1 denotes the multiplicative inverse of x modulo q. Salie evaluated R(d; q) in the classical setting n = 1 for even q, and for odd q = p with > 1. Later, Smith provided formulas that simplied the computation of R(d; q) in these cases for n > 1. Recently, Cochrane, Liu and Zheng computed upper bounds for R(d; q) in the general case n > 0, stopping short of their explicit evaluation. Here I complete the computation they initiated to obtain explicit values for the Kloosterman sums for > 1, relying on basic properties of some simple specialized exponential sums. The treatment here is more elementary than the author's previous determination of these Kloosterman sums using character theory and p-adic methods. At the least, it provides an alternative, independent evaluation of the Kloosterman sums. Mathematics Subject Classic ation 11L05, 11T24

Explicit values of multi-dimensional Kloosterman sums for prime powers, II