Unramified Solutions of Inverse Galois Problems and their Applications

首席研究员Nomura研究了循环立方场上非分枝3-扩展的存在性，给出了循环立方场的3-类场塔长度大于1的充分条件。野村还研究了g -扩展中分支的素数。设p是奇质数。Scholz和Reichardt证明了每一个p群G都可以被实现为有理数域Q的某个扩展M的伽罗瓦群。对于有限p群G，设t-ram(G)表示最小整数，使得G可以被实现为Q的一个有限分形扩展的伽罗瓦群，只在t-ram(G)有限素数处被分形。我们用d(G)表示有限p群G的最小生成子数，即我们熟知的d(G) t-ram(G)≦n，其中p^n是G的阶数。野村证明了plan结果的改进。Nomura也证明了t-ram(G) =d(G)对于任何小于或等于243阶的3群G。伊藤研究员以小组理论思考的方式参与了本研究。Hirabayashi和Kimura关于理想阶级群体的思考对本研究起到了重要作用。

Head investigator Nomura studied the existence of unramified 3-extensions over cyclic cubic fields and gave a sufficient condition that the length of the 3-class field tower of a cyclic cubic field is greater than 1.Nomura also studied the number of primes which are ramified in G-extension. Let p be an odd prime number. Scholz and Reichardt proved that every p-group G can be realized as the Galois group of some extension M of the rational number field Q. For a finite p-group G, let t-ram(G) denote the minimal integer such that G can be realized as the Galois group of a tamely ramified extension of Q ramified only at t-ram(G) finite primes. We denote by d(G) the minimal number of generators of a finite p-group G. Then it is well-known d(G) t-ram(G) ≦n, where p^n is the order of G. Plans(2004)proved a better upper bound for t-ram(G). Nomura proved an improvement of the result of Plans. Nomura also proved that t-ram(G) =d(G) for any 3-group G of order less than or equal to 243.Investigator Ito cooperated in this research by group theoretical consideration. And considerations by Hirabayashi and Kimura concerning the ideal class group played an important role to this research.