Kummer Extensions And Generalized Kummer Extensions

In this dissertation.we mainly discuss the Kummer theory of a field and a type of generalized Kummer extensions on function fields,and obtain some arithmetic of generalized Kummer extensions.In the first chapter,we introduce the Kummer theory of a field,what it reveals is that there is a correspondence between the abelian extensions L over K of exponent n and the groups△such that A(?)△(?)AK:△→L=K((?)-1(△)),L→△=A(?)∩AK.In the second chapter,we consider the extension on the function fields deter-mined by a special polynomial F(y)=yqn+an-1yqn+…+a2yq2+a1yq+ay. We name this extension generalized Kummer extension.Then we obtain some re-sults:(1)The set R of the roots of F(y)consists of a Fq vector space of dimension n.(2)Gal(L／K)is a subgroup of GLn(Fq).Because it's very diffcult to find a method to do research with the generalized Kummer extension in general,then this dissertation consider the extension determined L／K by the special polynomial F(y)=yq2+ayq+Thy,a∈(Th)when n=2.To this type extension,we have Gal(L／K)(?)GL2(Fq).The third chapter mainly consider the simple case of the special polynomial in last chapter when n=2,h=1:F(y)=yq2+ayq+Ty,a∈(T).We research the ramifications of prime places and calculate the genus of L and its some subfiled. The main results is:(1)the only places ramified in L／K are T and∞.(2)denote deg a=n,then the genus of L is (?)(n-1)+(?).