整体函数域素数次循环扩域的类群

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F_q(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.

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

Let K/F be a cyclic extension of prime degree l of global function fields, where F is finite separable extension of F_q(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.