Galois Groups And Genera Of A Kind Of Quasi-cyclotomic Function Fields

In number theory of function fields, the fields we most understand are quadratic function fields and cyclotomic function fields. There are rich theories about these two kinds of function fields, but there is no kinds of non-abelian function fields we under-stand well. Therefore, it is very meaningful to construct a kind of non-abelian function fields and study their arithmetic properties.We call a (q-1)-th Kummer extension of a cyclotomic function field a quasi-cyclotomic function field if it is Galois, but non-abelian, over the rational function field with the constant field of q elements. Although we haven't known all the quasi-cyclotomic function fields, a large kind of such fields are described explicitly in [1] following the works in [2] and in [3]. In this thesis, we determine the structure of the Galois groups of this kind of quasi-cyclotomic function fields over the base field. We also give the genus formulae of them.For each quasi-cyclotomic function field of this kind, first we decompose the Ga-lois group of the cyclotomic function field as the direct product of its p-Sylow sub-group and the other subgroup which is generated by finite generators, then we follow the method in [4] and use the results in [1] and in [3] to lift these finite generators to the Galois group of the quasi-cyclotomic function field, calculate the orders of these lift-ings and determine their relations. The Galois group of the quasi-cyclotomic function field can just be decomposed as the direct product of its p-Sylow subgroup and the subgroup generated by these liftings. To get the genus formula, first we get the genus formula for the cyclotomic function field by applying Riemann-Hurwitz theorem and a result in [5], then we use Hasse's genus formula on Kummer extensions to get the desired genus formula.