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Wikipedia / 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

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

since it may contain infinitely large elements over the latter.

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

of Gaussian rationals) is also formally 5-adic. On the other hand, the field of Gaussian rationals is not and its residue field has 9 elements.

-adic but that there does not exist any proper algebraic is said to be p-adically closed-adically closed, and so is the algebraic closure-adic algebraic numbers).

-adically closed, then

  • -adically closed),

  • is Henselian with respect to this place (that is, its valuation ring is so),

  • is exactly the image of the Kochen operator (see below),

  • ) 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.

such that

  • its characteristic),

  • ),

  • ),

-adically complete fields. These hypotheses are notably satisfied for the field of rationals, with the prime number having valuation 1.

The Kochen operator

satisfying the hypothesis and with the notations introduced in the previous paragraph, define the Kochen operator by

-adic) analogue of the square function in the real case.

-adic if and only if is not a sum of squares.

First-order theory

-adic valuation) is complete and model complete, and if we slightly enrich the language it admits quantifier elimination-adically closed fields as those whose first-order theory is elementarily equivalent.

Notes

References