Ab-faithfulness of absolute Galois groups in number theory

Source abstract

We prove that the absolute Galois group of every discretely valued field of mixed characteristic is ab-faithful. We also prove ab-faithfulness for higher local fields of positive dimension and give further examples using finite split embedding problems. Finally, we give a formal Hilbert~90 criterion for center-freeness and slimness of finite-step solvable quotients. We apply this criterion to ab-faithful absolute Galois groups and obtain slimness for several classes of fields, including higher local fields of positive dimension.