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 be the field of rational numbers and be its usual -adic valuation (with ). If is a (not necessarily algebraic) extension field of , itself equipped with a valuation , we say that is formally p-adic when the following conditions are satisfied:

• extends (that is, for all ),
• the residue field of coincides with the residue field of (the residue field being the quotient of the valuation ring by its maximal ideal ),
• the smallest positive value of coincides with the smallest positive value of (namely 1, since v was assumed to be normalized): in other words, a uniformizer for remains a uniformizer for .

(Note that the value group of K may be larger than that of F since it may contain infinitely large elements over the latter.)

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

For example, the field (i) of Gaussian rationals, if equipped with the valuation w given by (and ) is formally 5-adic (the place v=5 of the rationals splits in two places of the Gaussian rationals since 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 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:

• 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 , 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 (the value group of K) of a divisible group, with the lexicographical order.

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

• 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. ),
• K has finite absolute ramification, i.e., is finite (that is, a finite multiple of ),

(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 -adic if 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 ). It is easy to check that 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 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 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 .

Notes

References