Composite valuations and valuation topologies (tutorial)
by
Jochen Koenigsmann
University of Konstanz

The birth of valuation theory may be dated a good century ago when Hensel observed that one can bring topology into number theory not only by way of the usual (archimedean) absolute value but also via p-adic absolute values (one for each prime p), where two numbers are considered close when they are congruent modulo a high power of p. The p-adic metric thus induced on number fields allows new completions, the fields of p-adic numbers which are - in virtue of Hensel's Lemma - arithmetically as handy as the fields of real or complex numbers, and of equal importance to number theory.

This original topological flavour carries over to general (Krull) valuations of arbitrary rank and, indeed, there is a purely topological characterization of field topologies arising from valuations and absolute values. The tutorial gives an introduction to the well-known basic concepts and facts associated to valuation topologies:

Valuations inducing different topologies on a field (independent valuations) satisfy the so-called approximation theorem, while dependent valuations have a common coarsening, i.e. they correspond to the composition of one common place with distinct places on the residue field.

Generalizing a famous theorem of F.K.Schmitt, we also present the fact that any two henselian valuations on a field which is not separably closed are dependent, i.e. that there is at most one henselian topology on such fields. These henselian topologies will (more or less) be characterized as those field topologies for which the implicit function theorem holds. We shall also discuss the relation between henselization and completion.

Passing to coarsenings, refinements, compositions, completions and, of course, to henselizations of valued fields have now become standard techniques in valuation theory.

Date received: July 7, 1999