Applications of Weil’s Theorem on Character Sums

Notes for a talk given at LSBU on 7 September 2007 Finite fields Fq is the finite field of q elements, q a power of a prime p. Fq is the splitting field of X q − X over Fp. Fq is unique up to isomorphism. Let r = q k. Every automorphism of Fr over Fq is of the form x ↦ → x qi, i = 0, 1,..., k − 1. Thus the Galois group of Fr over Fq is cyclic with generator x ↦ → x q. The trace of an element x ∈ Fr is T (x) = x + x q + x q2 + · · · + xqk−1. Differential operator: D(a0 + a1x + · · · + anx n) = a1 + 2a2x + · · · + nanx n−1. Suppose K is a field of characteristic p, m ≤ p, a(x) is a polynomial in K[x], and for some x0 ∈ K