Model Theory of Valued Fields

We give a proposal for future development of the model theory of valued fields. We also summarize recent results on p-adic numbers. Let K be a valued field with a valuation map v : K → G ∪ {∞} to an ordered group 1 G; this is a map satisfying (i) v(x) = ∞ if and only if x = 0; (ii) v(xy) = v(x) + v(y) for all x, y ∈ K; (iii) v(x + y) ≥ min{v(x), v(y)} for all x, y ∈ K. We write R for the valuation ring {x ∈ K | v(x) ≥ 0} of K, M for its unique maximal ideal {x ∈ K | v(x) > 0} and we write k for the residue field R/M and ¯: R → k : x → ¯ x for the natural projection. We call K a Henselian valued field if R is a Henselian valuation ring. A valued field often carries an angular component map modulo M , or angular component map for short; it is a group homomorphism ac : K × → k × , extended by putting ac(0) = 0, and satisfying ac(x) = ¯ x for all x with v(x) = 0 (see [18]). 2. Different languages The model theory of valued fields can be studied at different levels of complexity. One can use the most basic language to study fields, the language L ring of rings, and for some valued fields (like the p-adic numbers) the relation v(x) ≤ v(y) is already definable in this language. In general, this relation is not automatically definable (like in algebraically closed valued fields). It is very natural to add a divisibility predicate, or even more conveniently, a restricted division function D as follows D : K 2 → K 2 : (x, y) → x/y if v(x) ≥ v(y), y = 0; 0 else. Let us call L D the language L ring together with D. Algebraically closed valued fields have quantifier elimination in L D , see [19]. It is also convenient to add unary predicates P n to L ring , corresponding to the set of n-th powers in K × = K \ {0}; one thus obtains the language of Macintyre L Mac. Macintyre [15] proved that p-adic fields have quantifier elimination 2 in L Mac. It is less known that there are many other valued fields which have …