Periodica Mathematica Hungarica | 2021
On the maximal unramified pro-2-extension of certain cyclotomic \n \n \n \n $$\\mathbb {Z}_2$$\n \n \n Z\n 2\n \n \n -extensions
Abstract
In this paper, we establish a necessary and sufficient criterion for a finite metabelian 2-group G whose abelianized $$G^{ab}$$ G ab is of type $$(2, 2^m)$$ ( 2 , 2 m ) , with $$m\\ge 2$$ m ≥ 2 , to be metacyclic. This criterion is based on the rank of the maximal subgroup of G which contains the three normal subgroups of G of index 4. Then, we apply this result to study the structure of the Galois group of the maximal unramified pro-2-extension of the cyclotomic $$\\mathbb {Z}_2$$ Z 2 -extension of certain number fields. Illustration is given by some real quadratic fields.