P-adically closed field explained

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.^[1]

Definition

Let

K be the field 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: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 .(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

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:^[2]

• 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

Z (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.

v(\pi) =1 ),

• K has finite absolute ramification, i.e.,

v(p) is finite (that is, a finite multiple of v(\pi) =1 ),(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 ak{p} -adic if ak{p} 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:

\gamma(z) = (1)/(\pi) (z ^{q-z})/((z ^{q-z)} ^{2-1}) (when z ^{q-z} ≠ \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

(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

Q _{p} .

Notes

1. Ax & Kochen (1965)
2. Jarden & Roquette (1980), lemma 4.1

References

• Ax . James . Kochen . Simon . Diophantine problems over local fields. II. A complete set of axioms for -adic number theory . Amer. J. Math. . 87 . 1965 . 631 - 648 . 10.2307/2373066 . 2373066 . 3 . The Johns Hopkins University Press .
• Kochen . Simon . Integer valued rational functions over the -adic numbers: A -adic analogue of the theory of real fields . 1969 . Number Theory (Proc. Sympos. Pure Math., Vol. XII, Houston, Tex., 1967) . 57 - 73 . American Mathematical Society .
  • Jarden . Moshe . Roquette . Peter . J. Math. Soc. Jpn. . 1980 . 32 . The Nullstellensatz over -adically closed fields . 425 - 460 . 10.2969/jmsj/03230425 . 3 . free .