Research of the inverse Galois problems with restricted ramifications and their applications to the class field tower problems

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The main purpose of this research is to study the inverse Galois problems with restricted ramifications for p-groups and their applications to the class field tower problems.Let p and q be distinct odd primes such that p-1 or p+1 is divisible by q. Let E be a non-abelian group of order p cubed, and let k be a cyclic extension over rational number field Q. We obtained the sufficient conditions for the existence of the unramified extension L/k such that the Galois group is isomorphic to E. By computing with PARI, we also gave some examples of cyclic fields which has an unramified extension with the Galois group E.