Title: On Galois extensions with cyclic decomposition groups
Abstract: Call a Galois extension of number fields "locally cyclic" if all its decomposition groups are cyclic.
Such extensions are interesting by themselves, but are also of significance for other problems in and around inverse Galois theory.
I will give a survey of several recent results by myself and others, in part on realizing certain nonsolvable groups (notably, symmetric groups, answering a question by Sonn and Bubboloni) as Galois groups of locally cyclic extensions of Q, in part on using such extensions for the solution of other number-theoretic problems such as construction of unramified extensions over low degree number fields.