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On some classes of Hilbertian fields. (English) Zbl 0547.12016
The author introduces a new and important class of fields K, called ”strongly pseudo real closed” (sprc), defined by the following property: Every field extension \(F| K\) which is at the same time totally real and totally transcendental, can be embedded over K into an elementary extension \({}^*K\) of K such that \({}^*K| F\) is totally transcendental. Recall that \(F| K\) is ”totally transcendental” (tt) if K is algebraically closed in F. Also, \(F| K\) is ”totally real” (tr) if every ordering of F can be extended to an ordering of K. [Warning: This terminology is not compatible with that of ”totally real number field” in the usual sense of algebraic number theory.]
The author’s concept of sprc field has been inspired, on the one hand by A. Prestel’s concept of pseudo real closed (prc) fields [Lect. Notes Math. 872, 127-156 (1981; Zbl 0466.12018)], and on the other hand by the Gilmore-Robinson characterization of Hilbertian fields [cf. the reviewer Lect. Notes Math. 498, 231-275 (1975; Zbl 0316.12103)]. Recall on the one hand, that K is prc if every tt and tr extension \(F| K\) can be K-embedded into an elementary extension \({}^*K\) of K. On the other hand, K is Hilbertian if Hilbert’s irreducibility theorem holds over K; by Gilmore-Robinson this is the case iff the field K(t) of rational functions in one variable can be embedded over K into an elementary extension \({}^*K\) of K such that \({}^*K\) is tt over K(t).
The object of this paper is to put into evidence that sprc fields do have interesting properties, combining in some way the properties of Prestel’s prc fields with those of Hilbertian fields. The class of sprc fields can be recursively axiomatized in the ordinary language L of fields, \(L=(0,1,+,-,\cdot).\) The non-real sprc fields are model complete with respect to a certain natural extension by definitions L’ of L. This extension is constructed in such a way that it admits as morphisms between fields precisely the tt extensions. Specifically, L’ is to contain new \((n+1)-\)ary predicates \(R_ n (n=1,2,...),\) and field theory is formulated in L’, besides by the usual field axioms, by adding a new axiom for each \(n=1,2,...\) which expresses the fact that \(R_ n(a_ 0,a_ 1,...a_ n)\) holds iff the polynomials \(a_ 0+a_ 1X+...+a_ nX^ n\) have a root in the field.
The field theory in L’ is model companionable, and the models of its model companion are precisely the non-real sprc fields. If we consider only fields of characteristic zero then the model companion is actually the model completion. The theory of non-real sprc fields of characteristic zero admits elimination of quantifiers, and it is decidable.
One section of the paper is devoted to the study of sprc fields with precisely n orderings, where n is a given natural number. It is shown that their theory can be developed quite analogously to Prestel’s theory of prc fields with precisely n orderings. But the question about elimination of quantifiers in this theory (with respect to a suitable extension of the language) had to be left open for \(n>0\). The author remarks in a note added in proof that a positive answer to this question has been given by Yu. L. Ershov [Dokl. Akad. Nauk SSSR, 266, No.3, 538-540 (1982), Prop. 5].
Special emphasis is given in this paper to ”Nullstellensätze” over a sprc field K; here K is supposed to be of characteristic zero. Let F be a function field over K, i.e. F is finitely generated over K. Suppose F is tt and tr over K. Consider the space \(\tilde S(F| K)=\tilde S\) of K- rational places of \(F| K\). Let \(\tilde H(F| K)=\tilde H\) denote the corresponding holomorphy ring, i.e. the intersection of the valuation rings of the places in \(\tilde S.\) If K is not algebraically closed then it is known that \(\tilde H\) is a Prüfer ring with F as its quotient field; the prime spectrum of \(\tilde H\) is in canonical bijection with \(\tilde H\). Moreover, and this seems to be of remarkable interest, \(\tilde H\) admits the following effective description: \(\tilde H\) is generated over K by elements of the form 1/z where either: \(z=g(y),\quad y\in F,\quad g(Y)\in K[Y]\) a monic polynomial with no root in K, or \(z=1+a_ 1x^ 2_ 1+...+a_ nx^ 2_ n\) with \(x_ 1,...,x_ n\in F\), \(n\geq 0\), and \(a_ 1,...,a_ n\in K\) such that there exists an ordering of K with \(a_ 1>0,...,a_ n>0\). Also, \(\tilde H\) is a so- called generalized Jacobson ring (i.e. the Jacobson radical of every finitely generated ideal of \(\tilde H\) coincides with its nilradical). From this, the ”Nullstellensätze” in traditional style, i.e. for polynomials in finitely many indeterminates with coefficients in the sprc field K, can be derived straightforwardly. All these facts are quite similar to the corresponding statements for p-adically closed fields.
This interesting paper which is beautifully written, contains a number of further important results which cannot all be mentioned here. It raises the question whether there exists a similar theory where not only orderings are studied but also p-valuations for a given finite set of primes p.
Remark: The list of literature as given in the paper should be enlarged to include at least the following titles which have appeared meanwhile: Yu. L. Ershov, Algebra Logika 22, No.4, 382-402 (1983); translated in Algebra Logic 22, 277-291 (1984; Zbl 0544.12017); M. Jarden, Acta Math. 150, 243-253 (1983; Zbl 0518.12018); B. Heinemann and A. Prestel, Fields regularly closed with respect to finitely many valuations and orderings, Quadratic and Hermitian Forms, Conf. Hamilton/Ont. 1983, CMS Conf. Proc. 4, 297-336 (1984); A. Prestel and the reviewer, Formally p-adic fields, Lect. Notes Math. 1050 (1984; Zbl 0523.12016); F.-V. Kuhlmann and A. Prestel, J. Reine Angew. Math. 353, 181-195 (1984; Zbl 0535.12015).
Reviewer: P.Roquette

MSC:
12F99 Field extensions
12L12 Model theory of fields
12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)
14A05 Relevant commutative algebra
03B25 Decidability of theories and sets of sentences
03C60 Model-theoretic algebra
12J15 Ordered fields
03C10 Quantifier elimination, model completeness, and related topics
11S99 Algebraic number theory: local fields
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
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References:
[1] J. Ax, Solving diophantine problems modulo every prime, Annals of Math. (2nd series) 85 (1967), 161–183. · Zbl 0239.10032
[2] J. Ax, The elementary theory of finite fields, Annals of Math. (2nd series) 88 (1968), 239–271. · Zbl 0195.05701
[3] S. Basarab, Definite functions on algebraic varieties over ordered fields, Revue Roumaine Math., XXIX (1984), No 7. · Zbl 0578.12019
[4] S. Basarab, Model complete theories of pseudo real closed fields, submitted to Revue Roumaine Math.
[5] S. Basarab, Extension of places and contraction properties for function fields over p-adically closed fields, J. reine angew. Math. 326 (1981), 54–78. · Zbl 0491.12025
[6] S. Basarab, A Nullstellensatz over ordered fields, Revue Roumaine Math., XXVIII (1983), No 7, 553–566. · Zbl 0538.14001
[7] S. Basarab, The Nullstellensatz over t-ordered fields: A t-adic analogue of the theory of formally p-adic fields, Preprint Series in Math., Bucuresti, No. 50 (1980). · Zbl 0451.12016
[8] S. Basarab, Axioms for pseudo real closed fields, Revue Roumaine Math., XXIX (1984), No. 6. · Zbl 0555.12009
[9] S. Basarab, V. Nica and D. Popescu, Approximation properties and existential completeness for ring morphisms, Manuscript a Math., vol. 33, fasc. 3/4 (1981), 227–228. · Zbl 0472.13013
[10] J. L. Bell and A. B. Slomson, Models and ultraproducts: An introduction, (1971) North-Holland. · Zbl 0179.31402
[11] G. Cherlin, Model theoretic algebra; selected topics, Lecture Notes Math. 521 (1976) Springer. · Zbl 0332.02056
[12] L. van den Dries, Model theory of fields, Thesis, Utrecht 1978.
[13] D. Dubois, A nullstellensatz for ordered fields, Arkiv für Matematik, 8 (1969), 111–114. · Zbl 0205.06102
[14] M. Jarden, Elementary statements over large algebraic fields, Trans. Amer. Math. Soc. 164 (1972), 67–91. · Zbl 0235.12104
[15] M. Jarden and P. Roquette, The Nullstellensatz over p-adically closed fields, J. Math. Soc. Japan, vol 32, No. 3 (1980), 425–460. · Zbl 0446.12016
[16] I. Kaplansky, Commutative rings, (1970) Allyn & Bacon, Boston. · Zbl 0203.34601
[17] C. Kiefe, Sets definable over finite fields: their zeta-function, Trans. Amer. Math. Soc. 223 (1976), 45–59. · Zbl 0372.02032
[18] W. Krull, Allgemeine Bewentungstheorie, J. reine angew. Math. 167 (1932), 160–196.
[19] K. McKenna, Pseudo henselian and pseudo real closed fields, Thesis, Yale.
[20] K. McKenna, Some diophantine Nullstellensätze, Lecture Notes Math. 834 (1980), Springer, 228–247. · Zbl 0452.13002
[21] K. McKenna, On the cohomology and elementary theory of a class of ordered fields, Preprint.
[22] M. Merckel, Wertbereiche rationaler funktionen, Diplomarbeit, Heidelberg 1975.
[23] A. Prestel, Pseudo real closed fields, Lecture Notes Math., 782 (1981), Springer, 127–156.
[24] A. Prestel and M. Ziegler, Model theoretic methods in the theory of topological fields, J. reine angew. Math. 229/300 (1978), 318–341. · Zbl 0367.12014
[25] A. Robinson, A relatively effective procedure for the solution of diophantine equations of positive genus, edited and supplemented by P. Roquette and G. Takeuti, Preprint.
[26] P. Roquette, Bemerkungen zur theorie der formal p-adischen Körper, Beiträge z. Algebra u. Geometrie, 1 (1971), 177–193. · Zbl 0245.12101
[27] P. Roquette, Principal ideal theorems for holomorphy rings in fields, J. reine angew. Math. 262/263 (1973), 361–374. · Zbl 0271.13009
[28] P. Roquette, A criterion for rational places over local fields, J. reine angew. Math. 292 (1977), 90–108. · Zbl 0346.14009
[29] P. Roquette, Nonstandard aspects of Hilbert’s irreducibility theorem, in Model theory and algebra. A memorial tribute to Abraham Robinson. Lecture Notes Math. 498 (1975), Springer, 231–275.
[30] G. Sacks, Saturated model theory (1972), W. A. Benjamin, Inc. · Zbl 0242.02054
[31] W. Schmidt, Equations over finite fields. An elementary approach, Lecture Notes Math. 536 (1976), Springer. · Zbl 0329.12001
[32] H. Schulting, Über reelle Stellen eines Körpers und ihren Holomorphiering, Dissertation, Dortmund 1979. · Zbl 0459.12014
[33] B. van der Waerden, Moderne Algebra I (1930), Springer. · JFM 56.0138.01
[34] A. Weil, Foundations of Algebraic Geometry, Amer, Math. Soc. Coll. Publ., 29 (1962). · Zbl 0168.18701
[35] W. Wheeler, Model complete theories of pseudo-algebraically closed fields, Preprint. · Zbl 0473.03029
[36] O. Zariski, Local uniformization on algebraic varieties, Annals Math. 41 (1940), 852–896. · JFM 66.1327.02
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