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函数域的双重根式扩域及素除子的密度
On the Double Radical Extensions of Algebraic Function Fields and Density of Prime Divisors
设K/Fq是整体函数域，l是与q互素的素数，ζl是K的固定代数闭包中的本原l次单位根. 对于a，b∈K*-（K*）l，本文主要讨论了根式扩域K（l√a）与K（l√a，l√b）的性质， 利用Kummer理论给出了K（l√a）/K与K（l√a，l√b）/K不是几何扩张的充要条件.当a，b是l-无关时， 对于K的素除子P及对应的离散赋值环OP， 利用这两类扩张的性质，通过分析a，b生成循环群（OP/P）*的充要条件，本文明确给出了满足使得a，b生成循环群（OP/P）*的全体素除子集合Ma，b的Dirichlet密度公式.
Let K/Fq be a global function field over the finite field Fq,and l be a prime number different from the characteristic of K. Denote by ζl a primitive l-th root of unity in a fixed algebraic closure of K. For two given elements a,b ∈ K*-(K*)l, we study in this paper the properties of radical extensions K(l√a) and K(l√a,l√b) of K. By the Kummer theory, we give a necessary and sufficient condition for K(l√a)/K and K(l√a,l√b)/K being not geometric extensions. Suppose that a,b ∈ K* - (K*)l are l-independent. For a prime divisor P of K and the corresponding discrete valuation ring OP, a necessary and sufficient condition for a,b,generating cyclic group (OP/P)* is presented by the properties of the above two function fields extensions. With the help of results obtained, the Dirichlet density of Ma,b, which is the set of prime divisors of K such that cyclic group (OP/P)* can be generated by a,b, is given explicitly in this paper.
函数域 / 根式扩张 / Kummer理论 / Dirichlet密度 {{custom_keyword}} /
function fields / radical extension / Kummer theory / Dirichlet density {{custom_keyword}} /
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国家自然科学基金资助项目（11601009）；安徽省自然科学基金资助项目（1608085QA04）
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