Gauss sums for general and special linear groups over a finite field

摘要

For a nontrivial additive character λ of the finite field with q elements and each positive integer r, the 'Gauss' sums λ((trg)r) over g GL(n,q) and over g SL(n,q) are considered. We show that both of them can be expressed as polynomials in q with coefficients involving certain exponential sums. As applications, we derive from these expressions the formulas for the number of elements g in GL (n,q) and SL (n,q) with g = p, for each p in the finite field with q elements.