Local Galois Theory In Dimension Two
This paper proves a generalization of Shafarevich's Conjecture, for fields of Laurent series in two variables over an arbitrary field. This result says that the absolute Galois group of such a field K is quasi-free of rank equal to the cardinality of K, i.e. every non-trivial finite split embedding problem for has exactly proper solutions. We also strengthen a result of Pop and Haran–Jarden on the existence of proper regular solutions to split embedding problems for curves over large fields; our strengthening concerns integral models of curves, which are two-dimensional.