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 $$\mathbb{Q}$$ of rational numbers and $$v$$ be its usual $p$-adic valuation (with $$v(p)=1$$). If $$F$$ is a (not necessarily algebraic) extension field of $$K$$, itself equipped with a valuation $$w$$, we say that $$(F,w)$$ is formally p-adic when the following conditions are satisfied: (Note that the value group of K may be larger than that of F since it may contain infinitely large elements over the latter.)
 * $$w$$ extends $$v$$ (that is, $$w(x)=v(x)$$ for all $$x\in K$$),
 * the residue field of $$w$$ coincides with the residue field of $$v$$ (the residue field being the quotient of the valuation ring $$\{x\in F : w(x)\geq 0\}$$ by its maximal ideal $$\{x\in F : w(x)>0\}$$),
 * the smallest positive value of $$w$$ coincides with the smallest positive value of $$v$$ (namely 1, since v was assumed to be normalized): in other words, a uniformizer for $$K$$ remains a uniformizer for $$F$$.

The formally p-adic fields can be viewed as an analogue of the formally real fields.

For example, the field $$\mathbb{Q}$$(i) of Gaussian rationals, if equipped with the valuation w given by $$w(2+i)=1$$ (and $$w(2-i)=0$$) is formally 5-adic (the place v=5 of the rationals splits in two places of the Gaussian rationals since $$X^2+1$$ 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 $$w(3)=1$$ 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.
 * there is a unique valuation w on F which makes F p-adically closed (so it is legitimate to say that F, rather than the pair $$(F,w)$$, is p-adically closed),
 * F is Henselian with respect to this place (that is, its valuation ring is so),
 * the valuation ring of F is exactly the image of the Kochen operator (see below),
 * the value group of F is an extension by $$\mathbb{Z}$$ (the value group of K) of a divisible group, with the lexicographical order.

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 $$\mathfrak{p}$$-adic if $$\mathfrak{p}$$ is the ideal corresponding to v) and v-adically complete fields.
 * the residue field of K is finite (call q its cardinal and p its characteristic),
 * the value group of v admits a smallest positive element (call it 1, and say π is a uniformizer, i.e. $$v(\pi)=1$$),
 * K has finite absolute ramification, i.e., $$v(p)$$ is finite (that is, a finite multiple of $$v(\pi)=1$$),

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:
 * $$\gamma(z) = \frac{1}{\pi}\,\frac{z^q-z}{(z^q-z)^2-1}$$

(when $$z^q-z \neq \pm 1$$). It is easy to check that $$\gamma(z)$$ 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 $$\frac{1}{\pi}$$ 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 $$-1$$ 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 $$\mathbb{Q}_p$$.