Unramified Solutions of Inverse Galois Problems and their Applications to the Class Field Tower Problems

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Head investigator Nomura studied the existence of unramified 3-extensions over cyclic cubic fields. In particular we treated the following problem.Problem P(F,G) : For a given Galois extension F/Q and a finite group G, does there exists an unramified Galois extension M/F such that the Galois group Gal(M/F) is isomorphic to G.Let F and K be the cyclic cubic fields satisfying the certain ramification conditions. Let G_1 and G_2 be the non-abelian 3-group such that the GAP-number is [81,9] and [243,2] respectively. One of main results is stated as follows.Assume that the class number of F is divisible by 81. Then, (1) there exists an unramified extension M/K such that the Galois group Gal(M/K) is isomorphic to G_1, (2) there exists an unramified extension M/F such that the Galois group Gal(M/F) is isomorphic to G_2.We also studied the class number relation between certain cubic fields, and gave an alternative proof of the Naito's result.Investigator Hirabayashi constructed some multiple Dedekind sums and gave a relative class number formula for an imaginary abelian number field by means of such Dedekind sums. He also gave a generalization of Girstmair's formula to an imaginary abelian number field.