函数域常值域扩张的类数整除性

On the Divisibility of Class Number in Constant Extensions of Algebraic Function Fields

设K/Fq是亏格大于0的整体函数域,Kn：=KF（qn）是K上的n次常值域扩张.利用整体函数域zeta函数的整系数多项式的有理表达式,结合函数域常值域扩张的基本性质,对于满足特定条件的素数l,本文讨论了使得除子类群Pic0（Kn）的Sylow-l子群为非平凡群的常值域扩张Kn的存在性.

Let K/Fq be a global function field over finite field Fq with genus greater than O. Suppose that Kn ：= KFqn is a constant field extension of K with degree n. Together the rational expression for zeta function of K with the properties of constant field extensions, for a specified prime number l, we study in this paper the existence of constant field extension Kn/K with 1 dividing the order of group Pic0（Kn）, which is the group of divisor classes of degree zero of function field Ks.