Abstract
Already Serre's "Cohomologie Galoisienne" contains an exercise regarding the following condition on a field $F$: For every finite field extension $E$ of $F$ and every $n$, the index of the $n$-th powers $(E^{*})^n$ in the multiplicative group $E^{*}$ is finite. Model theorists recently got interested in this condition, as it is satisfied by every superrosy field and also by every $\text{strongly}^2$ dependent field, and occurs in a conjecture of Shelah-Hasson on NIP fields. I will explain how it relates to the better known condition that $F$ is bounded (i.e. $F$ has only finitely many extensions of degree $n$, for any $n$ - in other words, the absolute Galois group of $F$ is a small profinite group) and why it is not preserved under elementary equivalence. Joint work with Franziska Jahnke.