Model theory of valued fields

A branch of model theory concerned with the elementary theories of fields with valuations $v$ (cf. Elementary theory; Field; Valuation). The basic first-order language is that of rings (or fields) together with a unary relation symbol for being an element of the valuation ring, or a binary relation symbol for valuation divisibility $v(x)\leq v(y)$ (cf. Structure).

Algebraically closed valued fields.

A. Robinson [a9] proved that the elementary theory of all algebraically closed valued fields is model complete. It can be deduced from his work that this theory is decidable and admits elimination of quantifiers. If, in addition, the characteristic of the fields and of their residue fields is fixed, then the theory so obtained is complete. The proof uses the fact that all extensions of a valuation of a field to its algebraic closure are conjugate, i.e., the algebraic closures of a valued field are isomorphic as valued fields. Robinson's results have witnessed many applications. The model completeness was applied to valued function fields (cf. Valued function field). The elimination of quantifiers was applied in the early 1990s by L. Lipshitz and H. Schoutens in different approaches towards a theory of rigid subanalytic sets, that is, subanalytic sets (cf. Semi-algebraic set) over an algebraically closed field that is complete under a valuation with Archimedean value group. The decidability is one ingredient in the proof that Hilbert's 10th problem has a positive solution for the ring of all algebraic integers (see Algebraic Diophantine equations).

Since the work of Robinson, model-theoretic results about various elementary theories of valued fields have been obtained. They can be understood as model-theoretic translations of a good structure theory for the fields in question, and in some cases they even challenged the development of such a structure theory. (See $p$-adically closed field; Real closed field.)

Model theory relative to value groups and residue fields.

J. Ax and S. Kochen [a1] used a model-theoretic result on ultraproducts of valued fields to prove a correct variant of Artin's conjecture (see Ultrafilter). The principle implicit in their result was stated explicitly by Yu. Ershov in [a5]: Assume that $(K,v)$ and $(L,w)$ are members of the class of all Henselian fields of residue characteristic $0$ (that is, with residue field of characteristic $0$). Then $(K,v)$ and $(L,w)$ are elementarily equivalent (in the first-order language of valued fields) if their value groups are elementarily equivalent (in the first-order language of ordered groups) and their residue fields are elementarily equivalent (in the first-order language of fields). That is, Henselian fields of residue characteristic $0$ can be classified up to elementary equivalence depending on the elementary theory of their residue fields and value groups.

This principle remains true if "elementary equivalence" is replaced by "elementary extension" (see Model theory) or by "existentially closed in" (cf. Existentially closed). A short proof of the latter version was given in [a7], where it was applied to study the Riemann spaces of algebraic function fields of characteristic $0$ (see Valued function field). Also, if the value group and the residue field have decidable elementary theories, then so does $(K,v)$. Such principles are called Ax–Kochen–Ershov principles. They have also been proved for other classes of valued fields.

For Henselian finitely ramified fields (which include the Henselian $p$-valued fields; see $p$-adically closed field), the "elementary extension" version of the Ax–Kochen–Ershov principle was proved by Ershov and, independently, by M. Ziegler in 1972. The same authors also settled the case of Kaplansky fields (cf. Kaplansky field) which are algebraically maximal, that is, do not admit non-trivial immediate algebraic extensions (see Valuation); since the Henselization of a valued field is an immediate algebraic extension, this condition yields that the fields in question are Henselian. The proofs of these principles implicitly use the fact that maximal immediate extensions of fields with residue fields of characteristic $0$, of finitely ramified fields and of Kaplansky fields are unique up to a valuation-preserving isomorphism. In the first two cases, this follows from the uniqueness of the Henselization, since the Henselizations of these fields are defectless fields (see Defect) and hence algebraically maximal; for the last case, see Kaplansky field. However, the uniqueness of maximal immediate extensions is not necessary for a class of valued fields to satisfy an Ax–Kochen–Ershov principle. Using a structure theory of the Henselizations of valued function fields (cf. Valued function field), the principle was extended to the class of tame fields (see Ramification theory of valued fields), cf. [a6]. This class contains all Henselian fields of residue characteristic $0$ and all algebraically maximal Kaplansky fields. But it also contains all algebraically maximal fields that are perfect of positive characteristic, and there are perfect valued fields for which the above uniqueness fails.

The corresponding "elementary equivalence" and "decidability" versions of the Ax–Kochen–Ershov principles hold for Henselian $p$-valued fields, for Henselian finitely ramified fields with $v(p)$ the smallest positive element of the value group, and for tame fields of fixed characteristic equal to that of the residue field. They do not carry over in general. For details, see [a6].

Relative elimination of quantifiers.

A direct translation of the above Ax–Kochen–Ershov principles to elimination of quantifiers yields an unsatisfactorily weak principle. Therefore, other concepts have been developed. V. Weispfenning [a10], extending work of P. Cohen (1969), A. Macintyre (1976) and many others, gave primitive-recursive algorithms for the elimination of all quantifiers up to those which refer to formulas about the elements of the value group, the residue field and certain other auxiliary structures like residue rings, using also Macintyre's power predicates (see $p$-adically closed field), which "hide" an existential quantifier. Weispfenning treated algebraically closed valued fields and several classes of Henselian fields of characteristic $0$. There are also results for Kaplansky fields. For a survey on the known results using this approach, see [a10].

One could also ask for criteria for an Ax–Kochen–Ershov principle expressing a relative version of substructure completeness (see Elimination of quantifiers). However, this already fails for the class of Henselian fields of residue characteristic $0$. But it works if "value group" and "residue field" are replaced by a stronger structure which extends the information contained in the value group and the residue field (S. Basarab (1991), F.-V. Kuhlmann (1994)). This structure can be used to classify algebraic extensions up to isomorphism. (Such classifications can help to create a link between substructure completeness and model completeness.) Modifications of this principle also cover the case of Henselian finitely ramified fields and of algebraically maximal Kaplansky fields. Macintyre's quantifier elimination for $p$-adically closed fields can be derived from the former. The general problem of relative elimination of quantifiers for the class of all tame fields is open.

Fields of Laurent series.

In [a2], Ax and Kochen considered the elementary theories of power series rings $K[[t]]$ and fields of formal Laurent series $K((t))$ over fields $K$ of characteristic $0$. Since they carry a canonical Henselian valuation with residue field $K$, it follows from the Ax–Kochen–Ershov principle for Henselian fields of residue characteristic $0$ that $K((t))$ and $L((t))$ are elementarily equivalent if and only if $K$ and $L$ are. The same holds for the rings $K[[t]]$ and $L[[t]]$. Ax and Kochen also showed that the elementary theories of $K[[t]]$ and of $K((t))$ are decidable if and only if the elementary theory of $K$ is decidable. In particular, the elementary theories of $\mathbf R[[t]]$ and of $\mathbf C[[t]]$ are decidable. They also showed that the ring of germs of analytic functions is elementarily equivalent to $\mathbf C[[t]]$ and hence its elementary theory is decidable. This contrasts with the undecidability of the ring of entire functions over $\mathbf C$, proved by R. Robinson in 1951.

It is not known (1996) whether the above results hold if $K$ has positive characteristic. In particular, it is not known whether the elementary theory of $\mathbf F_p((t))$ is model complete, decidable or admits elimination of quantifiers, where $\mathbf F_p$ denotes the field with $p$ elements. However, it is known that natural enrichments of the language of valued fields can lead to undecidability of the elementary theory of $\mathbf F_p((t))$ (results by J. Becker, J. Denef, L. Lipshitz (1979), and G.L. Cherlin (1982)). For more general results in this direction, see [a4].

There is a partial result which also holds for positive characteristic. If $K$ admits a Henselian valuation, then, in the language of fields, it is existentially closed in $K((t))$ (cf. [a6]). This shows that such a field $K$ is a large field in the sense of [a8], i.e., every smooth curve over $K$ has infinitely many $K$-rational points, provided it has at least one.

References

[a1]	J. Ax, S. Kochen, "Diophantine problems over local fields I" Amer. J. Math. , 87 (1965) pp. 605–630
[a2]	J. Ax, S. Kochen, "Diophantine problems over local fields III" Ann. of Math. , 83 (1966) pp. 437–456
[a3]	S.S. Brown, "Bounds on transfer principles for algebraically closed and complete valued fields" , Memoirs , 15 (204) , Amer. Math. Soc. (1978)
[a4]	F. Delon, Y. Rouani, "Indécidabilité de corps de séries formelles" J. Symb. Logic , 53 (1988) pp. 1227–1234
[a5]	Yu.L. Ershov, "On the elementary theory of maximal normed fields" Soviet Math. Dokl. , 6 (1965) pp. 1390–1393 (In Russian)
[a6]	F.-V. Kuhlmann, "Valuation theory of fields, abelian groups and modules" , Algebra, Logic and Applications , Gordon&Breach (to appear)
[a7]	F.-V. Kuhlmann, A. Prestel, "On places of algebraic function fields" J. Reine Angew. Math. , 353 (1984) pp. 181–195
[a8]	F. Pop, "Embedding problems over large fields" Ann. of Math. , 144 (1996) pp. 1–33
[a9]	A. Robinson, "Complete theories" , Amsterdam (1956)
[a10]	V. Weispfenning, "Quantifier elimination and decision procedures for valued fields" , Logic Colloquium Aachen 1983 , Lecture Notes in Mathematics , 1103 , Springer (1984) pp. 419–472
[a11]	W. Hodges, "Model theory" , Encycl. Math. Appl. , 42 , Cambridge Univ. Press (1993)