GAUSS SUMS ARE JUST CHARACTERS OF MULTIPLICATIVE GROUPS OF FINITE FIELDS 1

This paper is a summary of some papers [4,5,6] such that using special commutative group algebras, we could prove alternatively some reciprocity theorems, prime decom-positions of Gauss sums and Lenstra’s primality test. 1. Group Algebra Map(F,K) Let A = Map(F,K) be the set of all mappings from a finite field F = Fq of order q to a field K where q is a power of a prime p. Then we define the convolution product in A by the following (f ∗ g)(c) = ∑ a,b∈F a+b=c f(a)g(b) for f, g ∈ A and c ∈ F. This product together with the usual sum and the scalar product gives the structure of a commutative algebra over K. If there is no chance of confusion we shall denote the product f ∗ g by the usual notation fg. Let ua be the characteristic function of a ∈ F, namely, ua is defined by the following ua(b):= 1 if b = a 0 if b 6 = a. Then we have the following equations. uaub = ua+b and f = a∈F f(a)ua for f ∈ A. Thus {ua | a ∈ F} forms a basis of the group algebra A of the additive group of F over K. We denote by F ̂ the set of all characters of the multiplicative group F ∗ = F \ {0}, by χ[k] k-th power of χ ∈ F ̂ with respect to the convolution product and by the trivial character. We set (0) = 1 and χ(0) = 0 for χ 6 = ∈ F ̂. Thus we have F ̂ ⊂ A. We set J(f1, f2,..., fn) = (f1f2 · · · fn)(1) for f1, f2,..., fn ∈ A which i