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 GK 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 GK has exactly cardK 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.

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