Kloosterman sums and primitive elements in Galois fields
Stephen D. Cohen · Acta Arithmetica · 2000
1. Introduction. Let Fq denote the finite (Galois) field of order q, a power of a prime p. The multiplicative groupF q ofFq is cyclic of order q 1: a generator is known as a primitive element ofFq. HenceFq contains (q 1) primitive elements, where is Euler’s function. Generally, primitivity is a fragile property that may be destroyed when the element in question is modified through multiplication or addition. Nevertheless, if is a primitive element, then so is 1/ . When q = 2, H. Niederreiter [Ni] has expressed the number of irreducible polynomials of degreen ( 3) over the binary fieldF2, having the coecients of x n 1 and x both equal to 1, as a formula involving Kloosterman sums overF2 n. Thereby, this number is shown to be positive, except when n = 3. An alternative formulation of this conclusion is that, except when n = 3, F2n contains an element such that F2n = F2( ), and both and 1/ have (F2n,F2)-trace equal to 1. In this paper we consider extensions Fqn of a general finite field Fq. The aim is to show that Kloosterman sums are adequate for the stier task of generalising the above result (when