Construction of maximal unramified p-extensions with prescribed Galois groups

Abstract

In the present paper, we shall show that for any prime number p, every finite p-group occurs as the Galois Group of the maximal unramified p-extension over a certain number field of finite degree. We shall also show that for any given pro-p-group G with countably many generators, there exists a number field (not necessary of finite degree) whose maximal unramified p-extension has Galois group isomorphic to G. This means that the set of the isomorphism classes of the Galois groups of the maximal unramified p-extensions over the number fields (including of infinite degree) is precisely equal to that of all the pro-p-groups with countably many generators.

In this new version, a considerable improvement has been done, that is, Theorems 1 and 2 have been established for all the prime number p (without restrictive assumption C(p) on the prime number p)