Properties Of One Kind Of Linear Equations Over Finite Fields

Finite fields [2] are also called Galois fields. For any prime number p and a positive integer n, there exists one and only one finite field with pn elements, namely the splitting field of Xpn-X over Fp.It is well known about the general linear equations over finite fields [9]. Gauss [3] obtained some important results about quadratic congruences modulo a prime p. He also studied some special kinds of cubic and quartic congruence equations. Artin [1] gave his conjecture about the number of solutions of the equation y2=f(x) mod p which was proved by Hassc [6]. Later, Weil [12] proved a much more general theorem.In this paper, we study a special kind of linear equations whose variables belong to a subset of the finite field instead of the whole finite field. Let p>2 be a prime number, S C Fp.|S|=k,1≤κ≤p. Denote where Xi∈S. We have the following results:(1) The precise number of the solutions of F(x1,┄,xp-1)=b,b∈Fp for a special set S={0.1.┅,,κ-1} is obtained. Specially, a concrete explanation of Fermat's Little Theorem is provided. The distribution of the solutions of F(x1,┅,xp-1)=b.b∈Fp for all of the S with|S|=k is also studied.(2) A matrix is constructed using the solutions of f(r2,┅,xp-2)= b. b∈Fp and the properties of the matrix arc investigated, especially the relation between the eigenvalues of the matrix and the number of solutions.(3) An upper bound of the number of solutions of F((x1,┅,xp-1))= b,b∈Fp. f(x1,┅,xp-2)=b,b∈Fp is given by using Lev's theorem [7].This paper is organized as follows. Section 1 introduces some back-ground knowledge of finite fields and equations over finite fields. Section 2 describes some results about the solutions of F. Section 3 investigates the properties of the solutions of f. Section 4 introduces the Lev's theorem and its corollary.