Inverse Galois Problems with Restricted Ramifications

我们的研究成果总结如下。首席研究员野村研究了嵌入问题的非分支解和非分支p-扩张的存在性。当p是奇素数时，主要结果之一如下所述。设G是间隙数为[243，65]的群，它是243阶的非交换3-群。如果五次循环域F的类数可被3整除，则F上存在一个未分枝的Galois扩张使得Galois群与G同构。特别地，F的Hilbert类域的类数可被3整除。当p=2时，我们还研究了循环域上未分枝的四元数扩张的存在性，并肯定地回答了Fontaine-Mazur-Boston猜想的一个特例。他还利用改进的Milnor不变量给出了Redei符号的上同调解释，推广了Redei关于二次域的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.