P-adically closed field

A notion introduced by J. Ax and S. Kochen in [a1] and generalized by A. Prestel and P. Roquette in [a8]. A field $K$ of characteristic $0$ with a valuation $v$ is called finitely ramified if $K$ contains a prime element $\pi$ whose value $v(\pi)$ is a smallest positive element of the value group $vK$, and there are a prime number $p\in Z\subset K$ and a natural number $e$ such that $v(p)=e.v(\pi)$. In this case, the residue field has characteristic $p$ If, in addition, the residue field is finite, say, of cardinality $p^f$, then $(K,v)$ is called a $p$-valued field, and the natural number $e.f$ is called the $p$-rank of $(K,v)$. It is equal to the dimension of $\mathcal{O}/p\mathcal{O}$ as a vector space over the field with $p$ elements, where $\mathcal{O}$ denotes the valuation ring of $v$. The $p$-rank is $1$ if and only if $p$ is a prime element and the residue field is the field with $p$ elements; this is the case considered by Ax and Kochen.

A $p$-valued field is called $p$-adically closed if it does not admit any non-trivial $p$-valued algebraic extension of the same $p$-rank (cf. also Extension of a field). This holds if and only if $v$ is Henselian and $vK$ is a $Z$-group, that is, the quotient of $vK$ modulo the subgroup generated by the smallest positive element is a divisible group. The field $Q_p$ of $p$-adic numbers is $p$-adically closed of $p$-rank $1$, and the same holds for the relative algebraic closure of the field $Q$ of rational numbers in $Q_p$. There are also $p$-adically closed fields whose value group is not an 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 $d$ of $Q_p$ is $p$-adically closed of $p$-rank $d$. In the literature, $p$-valuations and $p$-adically closed fields are often tacitly assumed to have $p$-rank $1$.

In many respects, the theory of $p$-adically closed fields is similar to that of real closed fields (cf. Real closed field), see [a7].

Model theory.

Let $p$ be a fixed prime number. Ax and Kochen [a1] and Yu. Ershov [a4] showed that the elementary theory of all $p$-adically closed fields of $p$-rank $1$ is model complete and complete; hence, it is the same as the elementary theory of $Q_p$. 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 in the language of valued fields (cf. 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 $n\geq 2$, a power predicate $P_n$ interpreted by $P_n(x)\Leftrightarrow\exists y(y^n=x)$.

Prestel and Roquette [a8] generalized model completeness, elimination of quantifiers and decidability to the $p$-adically closed fields of fixed $p$>-rank, and completeness to the $p$-adically closed fields of fixed $p$-rank with prime element $p$.

Macintyre applied elimination of quantifiers to show that every infinite definable subset of a finite-dimensional $Q_p$-vector space has a non-empty interior. J. Denef gave another application in [a3]. See also [a7]; Elimination of quantifiers.

$p$-adic versions of Hilbert's 17th problem and Hilbert's Nullstellensatz.

An answer to Hilbert's 17th problem (see 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 $p$-adically closed fields, Kochen (1967) and Roquette (1971) proved a $p$-adic analogue which characterizes the rational functions $f\in K(X_1,...,X_n)$ over a $p$-adically closed field $K$ which are integral definite, that is, for every choice of $a_1,...,a_n\in K$, $f(a_1,...,a_n)$ lies in the valuation ring $\mathcal{O}$ of $K$ whenever it is defined. For $K$ of $p$-rank $1$ this characterization uses the Kochen operator

\begin{equation} \gamma(x)\frac{1}{p}\frac{x^p-x}{(x^p-x)^2-1}=\frac{1}{p}\bigg(p(x)-\frac{1}{p(x)}\bigg)^{-1}, \end{equation}

where $p(x)=x^p-x$ is the Artin–Schreier polynomial. Now $f$ is integral definite if and only if it is of the form

\begin{equation} f=\frac{\varphi}{1+p\psi} \end{equation}

with

\begin{equation} \varphi ,\psi\in Z[\gamma(K(X_1,...,X_n))] \end{equation}

The Kochen operator is the $p$-adic analogue of the square operator in the real case, and the above form is the analogue of the sum of squares.

The $p$-adic version of Hilbert's Nullstellensatz (cf. Hilbert theorem) reads as follows. Suppose that $f_1,...,f_m,g\in K[X_1,...,X_n]$. If $g$vanishes at all common roots of $f_1,...,f_k$ in $K^n$, then some power $g^N$ admits a representation $g^N=\lambda_1f_1+...+\lambda_mf_m$ with \lambda_i\in R.K[X_1,...,X_n], where $R$ consists of all quotients

\begin{equation} \frac{\varphi}{1+p\psi} \end{equation}

with

\begin{equation} \varphi ,\psi\in\mathcal{O}_K[\gamma(K(X_1,...,X_n))]. \end{equation}

The ring $R$ is called the Kochen ring of $K(X_1,...,X_n)$.

The Kochen operator and Kochen ring can also be used to characterize the formally $p$-adic fields, i.e., the fields admitting at least one $p$-valuation. For this and general versions of the above, see [a8].

Galois group of $Q_p$.

The (absolute) Galois group of $Q_p$ (that is, the Galois group of the algebraic closure of $Q_p$ over $Q_p$) was determined by U. Jannsen and K. Wingberg in [a5]. Like real closed fields, $p$-adically closed fields of $p$-rank $1$ are also characterized by their Galois group: any field with the same Galois group as $Q_p$ is $p$-adically closed of $p$-rank $1$. 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 $p\ne 2$. The full result was proved by J. Koenigsmann [a6], who gave a criterion for a field $K$ to admit a $p$>-Henselian valuation, that is, a valuation having a unique extension to the maximal Galois-$p$-extension of $K$, where $p$ is a prime number different from the characteristic of $K$. For $p=2$, this criterion was already given by R. Ware in 1981.

References

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