Inverse Galois Problems with Restricted Ramifications

我们的研究结果总结如下。首席研究员野村研究了嵌入问题的非分歧解和非分歧p-扩展的存在性。当p是奇素数时的一个主要结果如下。设G是GAP数为[243，65]的群，它是一个243阶的非交换3-群。若五次循环域F的类数可被3整除，则存在F上的非分歧伽罗瓦扩张，使得伽罗瓦群同构于G。特别地，F的希尔伯特类域的类数可以被3整除。在p=2的情形下，我们还研究了循环域上非分歧四元数扩张的存在性，并对Fontaine-Mazur-Boston猜想关于类域塔的Galois群的一个特例给出了肯定的回答。他还利用精化的Milnor不变量对Redei的符号给出了上同调的解释，推广了Redei关于二次域理想类群的一个经典结果。

Our research results are summarized as follows. Head investigator Nomura studied the unramified solution of embedding problems and the existence of unramified p-extensions. One of main results in case when p is an odd prime is stated as follows. Let G be the group such that the GAP-number is[243,65], which is a non-abelian 3-group of order 243. If the class number of quintic cyclic fields F is divisible by 3,then there exists an unramified Galois extension over F such that the Galois group is isomorphic to G. In particular, the class number of the Hilbert class field of F is divisible by 3. In case when p=2,we also studied the existence of unramified quaternion extension over cyclic fields, and gave an affirmative answer of a special case of Fontaine-Mazur-Boston conjecture concerning the Galois group of class field tower.Investigator Morishita discussed some analogies for primes coming from link theory, based on an analogy between the structure of the group of a link and those of certain Galois group. He also gave a cohomological interpretation of Redei's symbol by using refined Milnor invariants, and generalized a classical results of Redei concerning the ideal class group of quadratic fields.