Abstract. Weil's character sum estimate is used to study the problem of constructing generators for the multiplicative group of a finite field. An application to the distribution of irreducible polynomials is given, which confirms an asymptotic version of a conjecture of Hansen-Mullen.

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Table of ContentsChapter 1. Introduction 1.1. L-functions of exponential sums over finite fields 1.2. A brief history on the Newton polygon problem 1.3. Several decomposition theorems 1.4. Further remarks on open questions Chapter 2. A Lower Bound for the Newton Polygon 2.1. Non-degenerate Laurent polynomials 2.2. Definition of the Newton polygon 2.3. A lower bound: the Hodge polygon 2.4. The Adolphson-Sperber conjecture Chapter 3. Local Diagonal Theory 3.1. p-action, Gauss sums and L-functions 3.2. Applications of Stickelberger's theorem 3.3. Ordinary criterion and ordinary primes 3.4. Counter-examples in high dimensions Chapter 4. Global Decomposition Theory 4.1. Facial decomposition for the Newton polygon 4.2. Collapsing decomposition for generic Newton polygon 4.3. Hyperplane decomposition for generic Newton polygonThis paper is based on the author's series of lectures delivered at the January 1999 Mini-course in Number Theory, held at Sogang University (Seoul). The aim was to give an elementary and self-contained introduction to the theory of Newton polygons (namely, the p-adic Riemann hypothesis) for L-functions of exponential sums over a finite field. In addition to giving a thorough treatment of the basic elementary local theory, we also describe the deeper global theory and include some explicit examples to illustrate how to use the main theorems. In this way, it is hoped that these notes would at least give a little feeling and some indications about the subject of Newton polygons for L-functions. This arithmetic subject can be traced back

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