On The L-functions And Normal Bases Over Finite Fields

This thesis investigates the normal bases and L-functions over finite fields.In Chapter One, we introduce the exponential sums and L-functions over finite fields in section 1. The Newton polyhedron, Newton polygon, generical Newton polygon and the lower bound of Newton polygon-Hodge polygon are also introduced. In section 2, we introduce the normal bases of finite fields, and propose the main question:when do the set of normal polynomials of degree n equal to the set of monic irreducible polynomials of non-zero trace of degree n?In Chapter Two, we investigate the L-functions and exponential sums over finite fields. We investigate the L-functions and exponential sums of a family of Laurent polynomials F(a,b,c)(x, y, z) which appear in Ivaniec's work on Laplace-Beltrami operators acting on au-tomorphic functions with respect to the groupΓ0(p). We discuss the arithmetic property of zeros and poles of the L-functions of F(a,b,c)(x, y, z). We introduce Dwork's p-adic theory and Daqing Wan's two decomposition theorems, then we use them to get a Zariski open set which the Newton polynomial of corresponding L-function of F(a,b,c)(x, y, z) get its lower bound, the Zariski open set is the complement of a variety defined by a so called Hasse polynomial. Furthermore, we calculate the Hasse polynomial.In Chapter Three, we investigate the normal bases and normal polynomials. The set of normal polynomials of degree n are in the set of monic irreducible polynomials of degree n which have non-zero traces. It is interesting to ask when these two sets are equal. Perlis, Pei and Chang et al got partial answers respectively. Put their answers together, we have the answer of the question above is that the two sets equal if and only if n is a power of p where p is the character of field or n is a prime different from p and has q as one of its primitive roots. We will give a brief proof of this result based on two counting theorems.