Galois characterization of henselian fields
by
Jochen Koenigsmann
University of Konstanz

It was the discovery of Evariste Galois that arithmetic questions over a field K may be solved by studying the group of K-automorphims of algebraic extensions of K. As the recent work by Voevodsky, for example, illustrates, this approach is still highly attractive: deep arithmetic propteries of a field K are reflected in these automorphism groups, or to put it globally, in the absolute Galois gorup G_K of K, i.e. the Galois group of a separable algebraic closure of K over K. Artin-Schreier theory provides an easier example: K is formally real (resp. real closed) iff G_K contains an element (resp. is) of order 2. In this talk I shall show how non-trivial valuations of K (resp. non-trivial henselian valuations of K) are encoded in G_K.

Hilbert ramification theory describes G_K for a valued field K in terms of decomposition, inertia and ramification subgroup of G_K. Roughly speaking, we prove the converse: If G_K contains a subgroup Z which looks (group theoretically) like a decomposition group, K admits a valuation for which Z is a decomposition subgroup of G_K, so, in particular, the fixed field F of Z=G_F is henselian. To make this true one has to exclude certain trivial cases, e.g. that Z is projective or that inertia and ramification subgroup of Z coincide. Once these adjustments are made, however, the result can be proved in much stronger form, not using the full structure of decomposition groups, but only of a single Sylow-p-subgroup:

Theorem Let p be a prime, let F be a field and assume that the virtual p-th cohomological dimension of G_F is > 1. Then F admits a non-trivial henselian valuation with residual characteristic =/= p and with non-p-divisible value group iff any Sylow-p subgroup of G_F contains a non-trivial normal abelian subgroup.

As an application one can deduce the purely Galois-theoretic fact that solvable absolute Galois groups are metabelian.

Date received: July 7, 1999