The central problem in inverse Galois theory is to understand the structure of the absolute Galois group of a given field, such as $$extract_itex$$\mathbb{Q}$$/extract_itex$$. It is generally expected that significant information about $$extract_itex$$G_{\mathbb{Q}}$$/extract_itex$$ can be deduced from Galois theory over the local fields $$extract_itex$$\mathbb{Q}_p$$/extract_itex$$, which is much better understood. A guiding principle is to construct, or count extensions of $$extract_itex$$\mathbb{Q}$$/extract_itex$$ with a prescribed Galois group, and with prescribed local behaviour at some (finite or infinite!) set of primes. Specific instances of this approach lead to several problems of interest in number theory, such as minimal ramification problem, Grunwald problem, etc. In this talk I will focus on Galois realizations with conditions on the inertia subgroups.

I will discuss recent progress on two specific instances: Firstly, the construction of extensions with prescribed Galois group and "small" ramification indices. This is directly related to the construction of low-degree number fields with unramified $$extract_itex$$G$$/extract_itex$$-extensions and leads to generalized Cohen-Lenstra heuristics.

Secondly, the construction of $$extract_itex$$G$$/extract_itex$$-extensions with "powerfree" discriminant. This generalizes previous extensive investigations about fields with squarefree discriminants, corresponding to the special case $$extract_itex$$G=S_n$$/extract_itex$$.

If time allows, I will also discuss some work in progress about Galois realizations with prescribed decomposition groups.