Polynomials for Kloosterman Sums
by
Stan Gurak
University of San Diego

For fixed integer m with m > 1, the Kloosterman sums of order m are

R(d) = \sum_x \zeta_m^(x+dx*) ,

where 1 <= d <= m, (d, m)=1, \zeta_m = exp(2\pii/m) and x* denotes the multiplicative inverse of x modulo m. (The sum is over a complete system of reduced residues modulo m.) The Kloosterman sums satisfy a polynomial f_m(x) of degree \phi(m), which Salie (essentially) showed factors, when m=p is an odd prime, as a product of two distinct irreducible polynomials over the rationals, each of degree (p-1)/2.

Here I investigate the general case for composite m.

Date received: April 25, 2004