Construction of maximal unramified p-extensions with prescribed Galois groups

TL;DR

This paper demonstrates that any countably generated pro-p-group can be realized as the Galois group of the maximal unramified p-extension over some number field, covering both finite and infinite degree cases.

Contribution

It establishes the universality of pro-p-groups as Galois groups of maximal unramified p-extensions over appropriately constructed number fields.

Findings

Every finite p-group appears as a Galois group over some number field.

Any countably generated pro-p-group can be realized as a Galois group over a (possibly infinite degree) number field.

The set of all such Galois groups matches exactly the class of all pro-p-groups with countably many generators.

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.