We prove that every finite split embedding problem is solvable over the field K..X 1 ; : : : ; X n // of formal power series in n 2 variables over an arbitrary field K, as well as over the field Quot.AOEOEX 1 ; : : : ; X n / of formal power series in n 1 variables over a Noetherian integrally closed domain A. This generalizes a theorem of Harbater and Stevenson, who settled the case K..X 1 ; X 2 //.

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We extend the method of algebraic patching due to Haran-Jarden-Völklein from complete absolute valued fields to complete domains. We apply the extended method to reprove a result of Lefcourt obtained by formal patching -every finite group is regularly realizable over the quotient field of a complete domain.

We resolve an open problem in commutative algebra and Field Arithmetic, posed by Jarden. Let R be a generalized Krull domain. Is the ring RJ XK of formal power series over R a generalized Krull domain? We show that the answer is negative. Moreover, we show that essentially the opposite theorem holds. We prove that if R is a generalized Krull domain which is not a Krull domain, then RJ XK is never a generalized Krull domain.

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