View source for P-adically closed field

A notion introduced by J. Ax and S. Kochen in [[#References|[a1]]] and generalized by A. Prestel and P. Roquette in [[#References|[a8]]]. A [[Field|field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p1100102.png" /> of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p1100103.png" /> with a [[Valuation|valuation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p1100104.png" /> is called finitely ramified if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p1100105.png" /> contains a prime element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p1100106.png" /> whose value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p1100107.png" /> is a smallest positive element of the value group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p1100108.png" />, and there are a prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p1100109.png" /> and a natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001010.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001011.png" />. In this case, the residue field has characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001012.png" />. If, in addition, the residue field is finite, say, of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001013.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001014.png" /> is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001016.png" />-valued field, and the natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001017.png" /> is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001020.png" />-rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001021.png" />. It is equal to the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001022.png" /> as a [[Vector space|vector space]] over the field with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001023.png" /> elements, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001024.png" /> denotes the valuation ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001025.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001026.png" />-rank is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001027.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001028.png" /> is a prime element and the residue field is the field with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001029.png" /> elements; this is the case considered by Ax and Kochen.

A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001030.png" />-valued field is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001032.png" />-adically closed if it does not admit any non-trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001033.png" />-valued algebraic extension of the same <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001034.png" />-rank (cf. also [[Extension of a field|Extension of a field]]). This holds if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001035.png" /> is [[Henselian|Henselian]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001036.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001038.png" />-group, that is, the quotient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001039.png" /> modulo the subgroup generated by the smallest positive element is a [[Divisible group|divisible group]]. The field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001040.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001041.png" />-adic numbers is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001042.png" />-adically closed of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001043.png" />-rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001044.png" />, and the same holds for the relative algebraic closure of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001045.png" /> of rational numbers in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001046.png" />. There are also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001047.png" />-adically closed fields whose value group is not an [[Archimedean group|Archimedean group]]; they can be constructed by general valuation theory using the above criterion, but their existence can also be shown by a model-theoretic argument. Every extension of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001048.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001049.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001050.png" />-adically closed of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001051.png" />-rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001052.png" />. In the literature, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001053.png" />-valuations and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001054.png" />-adically closed fields are often tacitly assumed to have <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001055.png" />-rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001056.png" />.

In many respects, the theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001057.png" />-adically closed fields is similar to that of real closed fields (cf. [[Real closed field|Real closed field]]), see [[#References|[a7]]].

==Model theory.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001058.png" /> be a fixed prime number. Ax and Kochen [[#References|[a1]]] and Yu. Ershov [[#References|[a4]]] showed that the elementary theory of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001060.png" />-adically closed fields of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001061.png" />-rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001062.png" /> is model complete and complete; hence, it is the same as the elementary theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001063.png" />. Since a recursive set of axioms can be derived from the above characterization, it follows that their elementary theory is decidable. It does not admit [[Elimination of quantifiers|elimination of quantifiers]] in the language of valued fields (cf. [[Model theory of valued fields|Model theory of valued fields]]). However, in 1976 A. Macintyre showed that elimination of quantifiers can be obtained if one adjoins to the language, for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001064.png" />, a power predicate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001065.png" /> interpreted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001066.png" />.

Prestel and Roquette [[#References|[a8]]] generalized model completeness, elimination of quantifiers and decidability to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001067.png" />-adically closed fields of fixed [[P-rank|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001068.png" />-rank]], and completeness to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001069.png" />-adically closed fields of fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001070.png" />-rank with prime element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001071.png" />.

Macintyre applied elimination of quantifiers to show that every infinite definable subset of a finite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001072.png" />-vector space has a non-empty interior. J. Denef gave another application in [[#References|[a3]]]. See also [[#References|[a7]]]; [[Elimination of quantifiers|Elimination of quantifiers]].

==<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001073.png" />-adic versions of Hilbert's 17th problem and Hilbert's Nullstellensatz.==
An answer to Hilbert's 17th problem (see [[Real closed field|Real closed field]]) was given by E. Artin in 1927. A. Robinson reproved (in 1955) the result using the model completeness of real closed fields. Using the model completeness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001075.png" />-adically closed fields, Kochen (1967) and Roquette (1971) proved a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001076.png" />-adic analogue which characterizes the rational functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001077.png" /> over a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001078.png" />-adically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001079.png" /> which are integral definite, that is, for every choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001080.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001081.png" /> lies in the valuation ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001082.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001083.png" /> whenever it is defined. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001084.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001085.png" />-rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001086.png" /> this characterization uses the Kochen operator

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001087.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001088.png" /> is the Artin–Schreier polynomial. Now <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001089.png" /> is integral definite if and only if it is of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001090.png" /></td> </tr></table>

with

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001091.png" /></td> </tr></table>

The Kochen operator is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001092.png" />-adic analogue of the square operator in the real case, and the above form is the analogue of the sum of squares.

The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001094.png" />-adic version of Hilbert's Nullstellensatz (cf. [[Hilbert theorem|Hilbert theorem]]) reads as follows. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001095.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001096.png" /> vanishes at all common roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001097.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001098.png" />, then some power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p11001099.png" /> admits a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p110010100.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p110010101.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p110010102.png" /> consists of all quotients

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p110010103.png" /></td> </tr></table>

with

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p110010104.png" /></td> </tr></table>

The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p110010105.png" /> is called the Kochen ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p110010106.png" />.

The Kochen operator and Kochen ring can also be used to characterize the formally <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p110010108.png" />-adic fields, i.e., the fields admitting at least one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p110010109.png" />-valuation. For this and general versions of the above, see [[#References|[a8]]].

==Galois group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p110010110.png" />.==
The (absolute) Galois group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p110010112.png" /> (that is, the [[Galois group|Galois group]] of the algebraic closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p110010113.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p110010114.png" />) was determined by U. Jannsen and K. Wingberg in [[#References|[a5]]]. Like real closed fields, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p110010115.png" />-adically closed fields of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p110010116.png" />-rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p110010117.png" /> are also characterized by their Galois group: any field with the same Galois group as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p110010118.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p110010119.png" />-adically closed of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p110010120.png" />-rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p110010121.png" />. This was proved for fields of algebraic numbers by J. Neukirch in 1969, and by F. Pop in 1988 for Henselian fields with residue fields of positive characteristic. In 1995, I. Efrat proved the result for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p110010122.png" />. The full result was proved by J. Koenigsmann [[#References|[a6]]], who gave a criterion for a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p110010123.png" /> to admit a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p110010125.png" />-Henselian valuation, that is, a valuation having a unique extension to the maximal Galois-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p110010126.png" />-extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p110010127.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p110010128.png" /> is a prime number different from the characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p110010129.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110010/p110010130.png" />, this criterion was already given by R. Ware in 1981.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Ax,   S. Kochen,   "Diophantine problems over local fields II"  ''Amer. J. Math.'' , '''87'''  (1965)  pp. 631–648</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Cohen,   "Decision procedures for real and $p$-adic fields"  ''Comm. Pure and Appl. Math.'' , '''22'''  (1969)  pp. 131–151</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Denef,   "The rationality of the Poincaré series associated to the $p$-adic points on a variety"  ''Invent. Math.'' , '''77'''  (1984)  pp. 1–23</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  Yu.L. Ershov,   "On the elementary theory of maximal normed fields"  ''Soviet Math. Dokl.'' , '''6'''  (1965)  pp. 1390–1393  (In Russian)</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  U. Jannsen,   K. Wingberg,   "Die Struktur der absoluten Galoisgruppe $\mathfrak p$-adischer Zahlkörper"  ''Invent. Math.'' , '''70'''  (1982)  pp. 71–98 {{ZBL|0534.12010}}</TD></TR>
<TR><TD valign="top">[a6]</TD> <TD valign="top">  J. Koenigsmann,   "From $p$-rigid elements to valuations (with a Galois-characterization of $p$-adic fields)"  ''J. Reine Angew. Math.'' , '''465'''  (1995)  pp. 165–182</TD></TR>
<TR><TD valign="top">[a7]</TD> <TD valign="top">  A. Macintyre,   "Twenty years of $p$-adic model theory" , ''Logic Colloquium '84'' , Amsterdam  (1986)  pp. 121–153</TD></TR>
<TR><TD valign="top">[a8]</TD> <TD valign="top">  A. Prestel,   P. Roquette,   "Formally $p$-adic fields" , ''Lecture Notes in Mathematics'' , '''1050''' , Springer  (1984)</TD></TR>
</table>