Construction of maximal unramified p-extensions with prescribed Galois groups

Details Construction of maximal unramified p-extensions with prescribed Galois groups Construction of maximal unramified p-extensions with prescribed Galois groups

2007-05-16

arxiv.org/abs/0705.2293v2

Mathematics

Number Theory

In this new version, a considerable improvement has been done, that is, Theorems 1 and 2 have been established for all the pri

Scientific paper

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.

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