
P-adically closed field

In mathematics, a "p"-adically closed field is a field that enjoys a closure property that is a close analogue for "p"-adic fields to what real closure is to the real field. They were introduced by James Ax and Simon B. Kochen in 1965.
Definition.
Let "K" be the field ℚ of rational numbers and "v" be its usual "p"-adic valuation (with formula_1). If "F" is a (not necessarily algebraic) extension field of "K", itself equipped with a valuation "w", we say that formula_2 is formally "p"-adic when the following conditions are satisfied:
The formally "p"-adic fields can be viewed as an analogue of the formally real fields.
For example, the field ℚ(i) of Gaussian rationals, if equipped with the valuation w given by formula_6 (and formula_7) is formally 5-adic (the place "v"=5 of the rationals splits in two places of the Gaussian rationals since formula_8 factors over the residue field with 5 elements, and "w" is one of these places). The field of 5-adic numbers (which contains both the rationals and the Gaussian rationals embedded as per the place "w") is also formally 5-adic. On the other hand, the field of Gaussian rationals is "not" formally 3-adic for any valuation, because the only valuation "w" on it which extends the 3-adic valuation is given by formula_9 and its residue field has 9 elements.
When "F" is formally "p"-adic but that there does not exist any proper "algebraic" formally "p"-adic extension of "F", then "F" is said to be "p"-adically closed. For example, the field of "p"-adic numbers is "p"-adically closed, and so is the algebraic closure of the rationals inside it (the field of "p"-adic algebraic numbers).
If "F" is "p"-adically closed, then:
The first statement is an analogue of the fact that the order of a real-closed field is uniquely determined by the algebraic structure.
The definitions given above can be copied to a more general context: if "K" is a field equipped with a valuation "v" such that
(these hypotheses are satisfied for the field of rationals, with "q"=π="p" the prime number having valuation 1) then we can speak of formally "v"-adic fields (or formula_14-adic if formula_14 is the ideal corresponding to "v") and "v"-adically complete fields.
The Kochen operator.
If "K" is a field equipped with a valuation "v" satisfying the hypothesis and with the notations introduced in the previous paragraph, define the Kochen operator by:
(when formula_17). It is easy to check that formula_18 always has non-negative valuation. The Kochen operator can be thought of as a "p"-adic (or "v"-adic) analogue of the square function in the real case.
An extension field "F" of "K" is formally "v"-adic if and only if formula_19 does not belong to the subring generated over the value ring of "K" by the image of the Kochen operator on "F". This is an analogue of the statement (or definition) that a field is formally real when formula_20 is not a sum of squares.
First-order theory.
The first-order theory of "p"-adically closed fields (here we are restricting ourselves to the "p"-adic case, i.e., "K" is the field of rationals and "v" is the "p"-adic valuation) is complete and model complete, and if we slightly enrich the language it admits quantifier elimination. Thus, one can define "p"-adically closed fields as those whose first-order theory is elementarily equivalent to that of formula_21.