On some classes of Hilbertian fields.

*(English)*Zbl 0547.12016The 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).

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 |

##### Keywords:

recursively axiomatized class; pseudo real closed fields; strongly pseudo real closed; totally transcendental; totally real; Hilbertian fields; Hilbert’s irreducibility theorem; model complete; model companionable; elimination of quantifiers; decidable; orderings; Nullstellensätze; function field; holomorphy ring; Prüfer ring; generalized Jacobson ring; p-adically closed fields
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