Study Of The Arithmetic Of Local Fields

The basic object of the number theory is integer. Integer seems very easy. But it is very difficult to study it directly. To study it we can find some larger fields and embed ingter in them. The rational field is the first field that is discovered from real life. People think rational number can represent all in the world such as the length of line. As Pythagorean theorem is discovered, the first non-rational number 2^1/2 is found. People is obliged to exend the rational field to a larger field: real field. It is the first completion of rational number. The complex field is discovered for solving the polynomial equation. It is the completion of the algebraic field Q((-1)^1/2). People always work on the two fields in the following several hundred years. The other competion of Q is not found until the 20-th century. We call them local fields which are totally disconnected and different from the real field and the complex field.In mordern geometry, it is the well known method that the global property is obtained by the local analysis. In fact, the idea is very useful in number theory as well. All its' completions are local for an algebraic number field. For example, R and Q_p are local of Q. It is impossible to study the arithmetic question by purely local method. Howerer it is still very important to study the arithmetic properties of local fields. The connection of local data can show many global properties as well. The aim of the thesis is to study some properties of the local fields. Finally we work on the global field.By J. M. Fontain's theory, the local fields of characrteristic 0 have very import connection with those of characteristic p. He proved there is a natural isomorphism:where k is a perfect field of characteristic p, W(k) is the Witt ring of k, K is the fractional field of W(k), K|- is a fixed algebraic closure of K, K^cyc is the exension of K by joining all roots of unity. In this thesis, we will explain that the theory over characteristic 0 and that over characteristic p are closely related. For example, we must lift the object of characteristic p to that of characteristic 0 in the first chapter. In the second chapter, we need to use the above theorem to transform the problem of characteristic 0...