Split embedding problems over complete domains

Abstract

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 //.

Henselian implies large

“…The first point of this short note is to prove that actually K D k..x; y//, and more generally, K D Quot.R/ with R a complete Noetherian ring, are large fields, and that the class of large fields is much richer than previously believed. In particular, one can deduce Paran [Par09] from the already known fact that Problem 0 has a positive answer over large base fields K. Second, I give a lower bound for the number of distinct solutions of a nontrivial finite split embedding problem over a Hilbertian large field, a result which represents a wide extension of HarbaterStevenson [HS05]. Finally, using these results, one can generalize Harbater's result [Har09,Th.…”

“…1.1], by showing that every nontrivial finite split embedding problem for G K has jKj distinct proper solutions. And very recently, Paran [Par09] solved Problem 0 over K D Quot.R/, where R is a complete Noetherian local ring (satisfying some further technical conditions). The methods of proof in both cases are ingenious and quite technical.…”

“…The generalization of Paran [Par09] is the following: THEOREM 1.1. Let R be a domain which is Henselian with respect to some ideal a ¤ .0/.…”

Galois theory over complete local domains

“…By [15,Remark 2.4], R I 0 is a complete normed integral domain and the norm is given by (1), where b 0 , b in are now elements of B such that b in converges (with respect to | · |) to 0 as n → ∞. We consider the quotient rings We can now give the key proposition we need to control ramification.…”

“…We briefly recall the machinery used in [15,16] to solve embedding problems, and prove certain properties of the solution fields obtained by this machinery. …”