Let R be a domain, complete with respect to a norm which defines a non-discrete topology on R. We prove that the quotient field of R is ample, generalizing a theorem of Pop. We then consider the case where R is a ring of arithmetic power series which are holomorphic on the closed disc of radius 0<r<1 around the origin, and apply the above result to prove that the absolute Galois group of the quotient field of R is semi-free. This strengthens a theorem of Harbater, who solved the inverse Galois problem over these fields.
Bibliographical noteFunding Information:
The first author was supported by the European Commission under contract MRTN-CT-2006-035495. The second author was supported by an ERC grant.
Copyright 2011 Elsevier B.V., All rights reserved.
- Ample fields
- Galois theory
- Large fields
- Power series
- Semi-free profinite groups
- Split embedding problems