Unramified Solutions of Inverse Galois Problems and their Applications

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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.