Decidable regularly closed fields of algebraic numbers

1985 ◽  
Vol 50 (2) ◽  
pp. 468-475 ◽  
Author(s):  
Lou van den Dries ◽  
Rick L. Smith
Keyword(s):  
Galois Group ◽  
Elementary Theory ◽  
Algebraic Closure ◽  
Closed Field ◽  
Algebraic Numbers ◽  
Integer Polynomials ◽  
Forthcoming Book ◽  
Closed Fields ◽  
Free Profinite Group ◽  
Theoretic Argument

A field K is regularly closed if every absolutely irreducible affine variety defined over K has K-rational points. This notion was first isolated by Ax [A] in his work on the elementary theory of finite fields. Later Jarden [J2] and Jarden and Kiehne [JK] extended this in different directions. One of the primary results in this area is that the elementary properties of a regularly closed field K with a free Galois group (on either finitely or countably many generators) are determined by the set of integer polynomials in one indeterminate with a zero in K. The method of proof employed in [J1], [J2] and [JK] is unusual for algebra since it is a measure-theoretic argument. In this brief summary we have not made any attempt at completeness. We refer the reader to the recent paper of Cherlin, van den Dries, and Macintyre [CDM] and to the forthcoming book by Fried and Jarden [FJ] for a more thorough discussion of the latest results. We would like to thank Moshe Jarden, Angus Macintyre, and Zoe Chatzidakis for their comments on an earlier version of this paper.A countable field K is ω-free if the absolute Galois group , where is the algebraic closure of K and is the free profinite group on ℵ0 generators.

scholarly journals On the elementary theory of pairs of real closed fields. II

1982 ◽  
Vol 47 (3) ◽  
pp. 669-679 ◽  
Author(s):  
Walter Baur
Keyword(s):  
Power Series ◽  
Abelian Groups ◽  
Elementary Theory ◽  
Predicate Symbol ◽  
Axiom System ◽  
Closed Field ◽  
Algebraic Numbers ◽  
Real Closed Fields ◽  
Order Language ◽  
Closed Fields

Let ℒ be the first order language of field theory with an additional one place predicate symbol. In [B2] it was shown that the elementary theory T of the class of all pairs of real closed fields, i.e., ℒ-structures ‹K, L›, K a real closed field, L a real closed subfield of K, is undecidable.The aim of this paper is to show that the elementary theory Ts of a nontrivial subclass of containing many naturally occurring pairs of real closed fields is decidable (Theorem 3, §5). This result was announced in [B2]. An explicit axiom system for Ts will be given later. At this point let us just mention that any model of Ts, is elementarily equivalent to a pair of power series fields ‹R0((TA)), R1((TB))› where R0 is the field of real numbers, R1 = R0 or the field of real algebraic numbers, and B ⊆ A are ordered divisible abelian groups. Conversely, all these pairs of power series fields are models of Ts.Theorem 3 together with the undecidability result in [B2] answers some of the questions asked in Macintyre [M]. The proof of Theorem 3 uses the model theoretic techniques for valued fields introduced by Ax and Kochen [A-K] and Ershov [E] (see also [C-K]). The two main ingredients are(i) the completeness of the elementary theory of real closed fields with a distinguished dense proper real closed subfield (due to Robinson [R]),(ii) the decidability of the elementary theory of pairs of ordered divisible abelian groups (proved in §§1-4).I would like to thank Angus Macintyre for fruitful discussions concerning the subject. The valuation theoretic method of classifying theories of pairs of real closed fields is taken from [M].

scholarly journals DECIDABLE ALGEBRAIC FIELDS

2017 ◽  
Vol 82 (2) ◽  
pp. 474-488
Author(s):  
MOSHE JARDEN ◽  
ALEXANDRA SHLAPENTOKH
Keyword(s):  
Positive Integer ◽  
Elementary Theory ◽  
Algebraic Closure ◽  
Profinite Group ◽  
Algebraic Extension ◽  
Existential Theory ◽  
Free Profinite Group ◽  
Image Position ◽  
Algebraic Fields ◽  
Primitive Recursive

AbstractWe discuss the connection between decidability of a theory of a large algebraic extensions of ${\Bbb Q}$ and the recursiveness of the field as a subset of a fixed algebraic closure. In particular, we prove that if an algebraic extension K of ${\Bbb Q}$ has a decidable existential theory, then within any fixed algebraic closure $\widetilde{\Bbb Q}$ of ${\Bbb Q}$, the field K must be conjugate over ${\Bbb Q}$ to a field which is recursive as a subset of the algebraic closure. We also show that for each positive integer e there are infinitely many e-tuples $\sigma \in {\text{Gal}}\left( {\Bbb Q} \right)^e $ such that the field $\widetilde{\Bbb Q}\left( \sigma \right)$ is primitive recursive in $\widetilde{\Bbb Q}$ and its elementary theory is primitive recursively decidable. Moreover, $\widetilde{\Bbb Q}\left( \sigma \right)$ is PAC and ${\text{Gal}}\left( {\widetilde{\Bbb Q}\left( \sigma \right)} \right)$ is isomorphic to the free profinite group on e generators.

ON THE CLASSIFICATION OF CENTRALLY FINITE ALTERNATIVE DIVISION RINGS SATISFYING ALGEBRAIC CLOSURE CONDITIONS

2012 ◽  
Vol 11 (05) ◽  
pp. 1250088
Author(s):  
RICCARDO GHILONI
Keyword(s):  
Algebraic Closure ◽  
Real Closed Field ◽  
Closed Field ◽  
Real Closed Fields ◽  
Division Rings ◽  
Closed Fields ◽  
Closure Conditions ◽  
The One ◽  
Algebraically Closed Fields

In this paper, we prove that the rings of quaternions and of octonions over an arbitrary real closed field are algebraically closed in the sense of Eilenberg and Niven. As a consequence, we infer that some reasonable algebraic closure conditions, including the one of Eilenberg and Niven, are equivalent on the class of centrally finite alternative division rings. Furthermore, we classify centrally finite alternative division rings satisfying such equivalent algebraic closure conditions: up to isomorphism, they are either the algebraically closed fields or the rings of quaternions over real closed fields or the rings of octonions over real closed fields.

Independence of ℓ \ell -adic representations of geometric Galois groups

2018 ◽  
Vol 2018 (736) ◽  
pp. 69-93 ◽  
Author(s):  
Gebhard Böckle ◽  
Wojciech Gajda ◽  
Sebastian Petersen
Keyword(s):  
Galois Group ◽  
Finite Type ◽  
Algebraic Closure ◽  
Finite Extension ◽  
Field Extension ◽  
Galois Groups ◽  
Closed Field ◽  
Absolute Galois Group ◽  
Arbitrary Characteristic ◽  
Cohomology Modules

AbstractLetkbe an algebraically closed field of arbitrary characteristic, let{K/k}be a finitely generated field extension and letXbe a separated scheme of finite type overK. For each prime{\ell}, the absolute Galois group ofKacts on the{\ell}-adic étale cohomology modules ofX. We prove that this family of representations varying over{\ell}is almost independent in the sense of Serre, i.e., that the fixed fields inside an algebraic closure ofKof the kernels of the representations for all{\ell}become linearly disjoint over a finite extension ofK. In doing this, we also prove a number of interesting facts on the images and on the ramification of this family of representations.

The elementary theory of free pseudo p-adically closed fields of finite corank

1991 ◽  
Vol 56 (2) ◽  
pp. 484-496 ◽  
Author(s):  
Ido Efrat
Keyword(s):  
Rational Point ◽  
Elementary Theory ◽  
Valuation Rings ◽  
Order Language ◽  
Irreducible Variety ◽  
Closed Fields ◽  
Free Profinite Group ◽  
Unary Relation ◽  
Adic Valuation ◽  
Almost All

Let be p-adic closures of a countable Hilbertian field K. The main result of [EJ] asserts that the field has the following properties for almost all σ1,…,σe + m ϵ G(K) (in the sense of the unique Haar measure on G(K)e+m):(a) Kσ is pseudo p-adically closed (abbreviation: PpC), i.e., each nonempty absolutely irreducible variety defined over Kσ has a Kσ-rational point, provided that it has a simple rational point in each p-adic closure of Kσ.(b) G(Kσ) ≅ De,m, where De,m is the free profinite product of e copies Γ1,…, Γe of G(ℚp) and a free profinite group of rank m.(c) Kσ has exactly e nonequivalent p-adic valuation rings. They are the restrictions Oσ1,…, Oσe of the unique p-adic valuation rings on , respectively.In this paper we show that this result is in a certain sense the best possible. More precisely, we first show that the class of fields which satisfy (a)–(c) above is elementary in the appropriate language e(K), which is the ordinary first-order language of rings augmented by constant symbols for the elements of K and by e new unary relation symbols (interpreted as e p-adic valuation rings).

scholarly journals Maximal subfields of algebraically closed fields

1980 ◽  
Vol 29 (4) ◽  
pp. 462-468 ◽  
Author(s):  
Robert M. Guralnick ◽  
Michael D. Miller
Keyword(s):  
Galois Group ◽  
Characteristic Zero ◽  
Nonempty Subset ◽  
Rational Numbers ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Closed Fields ◽  
Algebraically Closed Fields

AbstractLet K be an algebraically closed field of characteristic zero, and S a nonempty subset of K such that S Q = Ø and card S < card K, where Q is the field of rational numbers. By Zorn's Lemma, there exist subfields F of K which are maximal with respect to the property of being disjoint from S. This paper examines such subfields and investigates the Galois group Gal K/F along with the lattice of intermediate subfields.

Effective Galois theory

1981 ◽  
Vol 46 (2) ◽  
pp. 385-392 ◽  
Author(s):  
Peter La Roche
Keyword(s):  
Galois Group ◽  
Galois Theory ◽  
Galois Groups ◽  
Profinite Group ◽  
Classical Literature ◽  
Profinite Groups ◽  
Algebraic Numbers ◽  
Recursively Enumerable ◽  
Free Profinite Group ◽  
Infinite Degree

Krull [4] extended Galois theory to arbitrary normal extensions, in which the Galois groups are precisely the profinite groups (i.e. totally disconnected, compact, Hausdorff groups). Metakides and Nerode [7] produced two recursively presented algebraic extensionsK⊂Fof the rationals such thatFis abelian,Fis of infinite degree overK, and the Galois group ofFoverK, although of cardinalityc, has only one recursive element (viz. the identity). This indicated the limits of effectiveness for Krull's theory. (The Galois theory offiniteextensions is completely effective.) Nerode suggested developing a natural effective version of Krull's theory (done here in §1).It is evident from the classical literature that the free profinite group on denumerably many generators can be obtained effectively as the Galois group of a recursive extension of the rationals over a subfield. Nerode conjectured that it could be obtained effectively as the Galois group of the algebraic numbers over a suitable subfield (done here in §2). The case of finitely many generators was done non-effectively by Jarden [3]. The author believes that the denumerable case, as presented in §2, is also new classically. Using this result and the effective Krull theory, every “co-recursively enumerable” profinite group is effectively the Galois group of a recursively enumerable field of algebraic numbers over a recursive subfield.

RATIONAL JORDAN DECOMPOSITION OF BILINEAR FORMS

2005 ◽  
Vol 07 (06) ◽  
pp. 769-786 ◽  
Author(s):  
DRAGOMIR Ž. ĐOKOVIĆ ◽  
KAIMING ZHAO
Keyword(s):  
Vector Space ◽  
Algebraic Closure ◽  
Real Field ◽  
Closed Field ◽  
Bilinear Forms ◽  
Dimensional Vector ◽  
Jordan Decomposition ◽  
Dimensional Vector Space ◽  
Closed Fields ◽  
Algebraically Closed Fields

This is a continuation of our previous work on Jordan decomposition of bilinear forms over algebraically closed fields of characteristic 0. In this note, we study Jordan decomposition of bilinear forms over any field K0 of characteristic 0. Let V0 be an n-dimensional vector space over K0. Denote by [Formula: see text] the space of bilinear forms f : V0 × V0 → K0. We say that f = g + h, where f, g, [Formula: see text], is a rational Jordan decomposition of f if, after extending the field K0 to an algebraic closure K, we obtain a Jordan decomposition over K. By using the Galois group of K/K0, we prove the existence of rational Jordan decompositions and describe a method for constructing all such decompositions. Several illustrative examples of rational Jordan decompositions of bilinear forms are included. We also show how to classify the unimodular congruence classes of bilinear forms over an algebraically closed field of characteristic different from 2 and over the real field.

Functorial properties of algebraic closure and Skolemization

1981 ◽  
Vol 31 (2) ◽  
pp. 136-141 ◽  
Author(s):  
C. J. Ash ◽  
A. Nerode
Keyword(s):  
Model Theory ◽  
Algebraic Closure ◽  
Closed Field ◽  
Ordered Sets ◽  
Algebraically Closed Field ◽  
Closed Fields ◽  
Algebraically Closed Fields

AbstractIt is shown that no functor F exists from the category of sets with injections, to the category of algebraically closed fields of given characteristic, with monomorphisms, having the properties that for all sets A. F(A) is an algebraically closed field having transcendence base A and for all injections f. F(f) extends f. There does exist such a functor from the category of linearly-ordered sets with order monomorphisms.An application to model-theory using the same methods is given showing that while the theory of algebraically closed fields is ω-stable, its Skolemization is not stable in any power.

scholarly journals Intersections of Real Closed Fields

1980 ◽  
Vol 32 (2) ◽  
pp. 431-440 ◽  
Author(s):  
Thomas C. Craven
Keyword(s):  
Galois Group ◽  
General Class ◽  
Entire Functions ◽  
Algebraic Closure ◽  
Careful Study ◽  
Real Numbers ◽  
The Real ◽  
Real Closed Fields ◽  
Closed Fields ◽  
Infinite Sequences

In this paper we wish to study fields which can be written as intersections of real closed fields. Several more restrictive classes of fields have received careful study (real closed fields by Artin and Schreier, hereditarily euclidean fields by Prestel and Ziegler [8], hereditarily Pythagorean fields by Becker [1]), with this more general class of fields sometimes mentioned in passing. We shall give several characterizations of this class in the next two sections. In § 2 we will be concerned with Gal , the Galois group of an algebraic closure F over F. We also relate the fields to the existence of multiplier sequences; these are infinite sequences of elements from the field which have nice properties with respect to certain sets of polynomials. For the real numbers, they are related to entire functions; generalizations can be found in [3].

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