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| {{DISPLAYTITLE:''p''-adically closed field}}
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| In [[mathematics]], a '''''p''-adically closed field''' is a [[Field (mathematics)|field]] that enjoys a closure property that is a close analogue for [[p-adic number|''p''-adic fields]] to what [[Real closed field|real closure]] is to the [[Real number|real field]]. They were introduced by [[James Ax]] and [[Simon B. Kochen]] in 1965.<ref>Ax & Kochen (1965)</ref>
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| == Definition ==
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| Let ''K'' be the field ℚ of [[rational number]]s and ''v'' be its usual [[p-adic order|''p''-adic]] [[Valuation (algebra)|valuation]] (with <math>v(p)=1</math>). If ''F'' is a (not necessarily algebraic) [[Field extension|extension field]] of ''K'', itself equipped with a valuation ''w'', we say that <math>(F,w)</math> is '''formally ''p''-adic''' when the following conditions are satisfied:
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| * ''w'' extends ''v'' (that is, <math>w(x)=v(x)</math> for all ''x'' in ''K''),
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| * the [[residue field]] of ''w'' coincides with the [[residue field]] of ''v'' (the residue field being the quotient of the valuation ring <math>\{x\in F : w(x)\geq 0\}</math> by its [[maximal ideal]] <math>\{x\in F : w(x)>0\}</math>),
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| * 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''.
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| (Note that the value group of ''K'' may be larger than that of ''F'' since it may contain infinitely large elements over the latter.)
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| The formally ''p''-adic fields can be viewed as an analogue of the formally real fields.
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| For example, the field ℚ(i) of [[Gaussian rational]]s, if equipped with the valuation w given by <math>w(2+i)=1</math> (and <math>w(2-i)=0</math>) is formally 5-adic (the place ''v''=5 of the rationals splits in two places of the Gaussian rationals since <math>X^2+1</math> 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 <math>w(3)=1</math> and its residue field has 9 elements.
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| 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).
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| If ''F'' is ''p''-adically closed, then:<ref>Jarden & Roquette (1980), lemma 4.1</ref>
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| * 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 <math>(F,w)</math>, is ''p''-adically closed),
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| * ''F'' is [[Henselian ring|Henselian]] with respect to this place (that is, its valuation ring is so),
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| * the valuation ring of ''F'' is exactly the image of the Kochen operator (see [[#The Kochen operator|below]]),
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| * the value group of ''F'' is an extension by ℤ (the value group of ''K'') of a divisible group, with the [[lexicographical order]].
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| The first statement is an analogue of the fact that the order of a real-closed field is uniquely determined by the algebraic structure.
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| The definitions given above can be copied to a more general context: if ''K'' is a field equipped with a valuation ''v'' such that
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| * the residue field of ''K'' is finite (call ''q'' its cardinal and ''p'' its characteristic),
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| * the value group of ''v'' admits a smallest positive element (call it 1, and say π is a uniformizer, i.e. <math>v(\pi)=1</math>),
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| * ''K'' has finite absolute ramification, i.e., <math>v(p)</math> is finite (that is, a finite multiple of <math>v(\pi)=1</math>),
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| (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 <math>\mathfrak{p}</math>-adic if <math>\mathfrak{p}</math> is the ideal corresponding to ''v'') and ''v''-adically complete fields.
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| == The Kochen operator ==
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| 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:
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| :<math>\gamma(z) = \frac{1}{\pi}\,\frac{z^q-z}{(z^q-z)^2-1}</math> | |
| (when <math>z^q-z \neq \pm 1</math>). It is easy to check that <math>\gamma(z)</math> 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.
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| An extension field ''F'' of ''K'' is formally ''v''-adic if and only if <math>\frac{1}{\pi}</math> 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 <math>-1</math> is not a sum of squares.
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| == First-order theory ==
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| 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 theory|complete]] and [[Model complete theory|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 <math>\mathbb{Q}_p</math>.
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| == References ==
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| * {{Cite journal | last1=Ax | first1=James | last2=Kochen | first2=Simon | title=Diophantine problems over local fields. II. A complete set of axioms for 𝑝-adic number theory | journal=Amer. J. Math. | volume=87 | year=1965 | pages=631–648 | doi=10.2307/2373066 | jstor=2373066 | issue=3 | publisher=The Johns Hopkins University Press }}
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| * {{Cite conference | last=Kochen | first=Simon | title=Integer valued rational functions over the 𝑝-adic numbers: A 𝑝-adic analogue of the theory of real fields | year=1969 | booktitle=Number Theory (Proc. Sympos. Pure Math., Vol. XII, Houston, Tex., 1967) | pages=57–73 | publisher=American Mathematical Society }}
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| * {{Cite web | url=http://eom.springer.de/P/p110010.htm | title=𝑝-adically closed field | last=Kuhlmann | first=F.-V. | publisher=[[Springer-Verlag]] | work=Springer Online Reference Works: Encyclopaedia of Mathematics | accessdate=2009-02-03 }}
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| * {{Cite journal | last1=Jarden | first1=Moshe | last2=Roquette | first2=Peter | journal=J. Math. Soc. Japan | year=1980 | volume=32 | title=The Nullstellensatz over 𝔭-adically closed fields | pages=425–460 | doi=10.2969/jmsj/03230425 | issue=3 }}
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| == Notes ==
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| <references />
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| [[Category:Field theory]]
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