整体函数域素数次循环扩域的类群

The Class Groups of Cyclic Extensions of Global Function Fields of Prime Degree

设K/F是整体函数域的素数l次循环扩张,F是有理函数域Fq（T）上的有限可分扩域.利用函数域的Conner-Hurrelbrink正合六边形与源于短正合列的正合六边形,本文在l整除与不整除基域F的理想类数的情形下,分别研究函数域K理想类群的Sylow l-子群的结构.同时,利用得到的结果,本文给出了基域F的单位为K中元素norm的若干条件.

Let K/F be a cyclic extension of prime degree l of global function fields, where F is finite separable extension of Fq（T）. Using Conner and Hurrelbrink＇s exact hexagon for function fields and other exact hexagon from short exact sequences, we study in this paper the structure of Sylow l-subgroup of the ideal class group of K. To be clear, we consider separately the two possibilities, when l divides the ideal class number of F and when it does not. Meanwhile, we summarize in this paper several conditions that characterize when every unit of F is norm of an element of the field K.