Note on the hilbert 2-class field tower

Let k be a number field with a 2-class group isomorphic to the Klein four-group. The aim of this paper is to give a characterization of capitulation types using group properties. Furthermore, as applications, we determine the structure of the second 2-class groups of some special Dirichlet fields k = Q(root d, root-1), which leads to a correction of some parts in the main results of A. Azizi and A. Zekhini (2020).