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].

scholarly journals A question of van den Dries and a theorem of Lipshitz and Robinson; Not everything is standard

2007 ◽  
Vol 72 (1) ◽  
pp. 119-122 ◽  
Author(s):  
Ehud Hrushovski ◽  
Ya'acov Peterzil
Keyword(s):  
Real Numbers ◽  
The Real ◽  
First Order ◽  
Real Closed Fields ◽  
Minimal Structure ◽  
Closed Fields ◽  
New Construction ◽  
The Relationship

AbstractWe use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions of real closed fields and structures over the real numbers. We write a first order sentence which is true in the Lipshitz-Robinson structure but fails in any possible interpretation over the field of real numbers.

Relative randomness and real closed fields

2005 ◽  
Vol 70 (1) ◽  
pp. 319-330 ◽  
Author(s):  
Alexander Raichev
Keyword(s):  
Real Number ◽  
Real Closed Field ◽  
Closed Field ◽  
Real Numbers ◽  
The Real ◽  
Real Closed Fields ◽  
Ordered Field ◽  
Closed Fields ◽  
Master's Thesis ◽  
National University

AbstractWe show that for any real number, the class of real numbers less random than it, in the sense of rK-reducibility, forms a countable real closed subfield of the real ordered field. This generalizes the well-known fact that the computable reals form a real closed field.With the same technique we show that the class of differences of computably enumerable reals (d.c.e. reals) and the class of computably approximable reals (c.a. reals) form real closed fields. The d.c.e. result was also proved nearly simultaneously and independently by Ng (Keng Meng Ng, Master's Thesis, National University of Singapore, in preparation).Lastly, we show that the class of d.c.e. reals is properly contained in the class or reals less random than Ω (the halting probability), which in turn is properly contained in the class of c.a. reals, and that neither the first nor last class is a randomness class (as captured by rK-reducibility).

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.

scholarly journals On Roots of Polynomials and Algebraically Closed Fields

2017 ◽  
Vol 25 (3) ◽  
pp. 185-195 ◽  
Author(s):  
Christoph Schwarzweller
Keyword(s):  
Algebraic Theory ◽  
Polynomial Rings ◽  
Multiple Roots ◽  
Real Numbers ◽  
The Real ◽  
Closed Fields ◽  
Roots Of Polynomials ◽  
Algebraically Closed Fields ◽  
Finite Domains ◽  
Identity Theorem

Summary In this article we further extend the algebraic theory of polynomial rings in Mizar [1, 2, 3]. We deal with roots and multiple roots of polynomials and show that both the real numbers and finite domains are not algebraically closed [5, 7]. We also prove the identity theorem for polynomials and that the number of multiple roots is bounded by the polynomial’s degree [4, 6].

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 Models of true arithmetic are integer parts of models of real exponentation

2021 ◽  
Vol 13 ◽  
Author(s):  
Merlin Carl ◽  
Lothar Sebastian Krapp
Keyword(s):  
Multiplicative Group ◽  
Peano Arithmetic ◽  
Real Closed Field ◽  
Closed Field ◽  
Integer Part ◽  
Real Numbers ◽  
The Real ◽  
Positive Elements ◽  
Closed Fields ◽  
Schanuel's Conjecture

Exploring further the connection between exponentiation on real closed fields and the existence of an integer part modelling strong fragments of arithmetic, we demonstrate that each model of true arithmetic is an integer part of an exponential real closed field that is elementarily equivalent to the real numbers with exponentiation and that each model of Peano arithmetic is an integer part of a real closed field that admits an isomorphism between its ordered additive and its ordered multiplicative group of positive elements. Under the assumption of Schanuel’s Conjecture, we obtain further strengthenings for the last statement.

scholarly journals Real weights, bound states and duality orbits

2016 ◽  
Vol 31 (01) ◽  
pp. 1550218 ◽  
Author(s):  
Alessio Marrani ◽  
Fabio Riccioni ◽  
Luca Romano
Keyword(s):  
Symmetry Group ◽  
Bound States ◽  
Specific Weight ◽  
The Other ◽  
Real Form ◽  
Real Numbers ◽  
The Real ◽  
Duality Symmetry ◽  
Infinite Sequences ◽  
Extremal Black Holes

We show that the duality orbits of extremal black holes in supergravity theories with symmetric scalar manifolds can be derived by studying the stabilizing subalgebras of suitable representatives, realized as bound states of specific weight vectors of the corresponding representation of the duality symmetry group. The weight vectors always correspond to weights that are real, where the reality properties are derived from the Tits–Satake diagram that identifies the real form of the Lie algebra of the duality symmetry group. Both [Formula: see text] magic Maxwell–Einstein supergravities and the semisimple infinite sequences of [Formula: see text] and [Formula: see text] theories in [Formula: see text] and [Formula: see text] are considered, and various results, obtained over the years in the literature using different methods, are retrieved. In particular, we show that the stratification of the orbits of these theories occurs because of very specific properties of the representations: in the case of the theory based on the real numbers, whose symmetry group is maximally noncompact and therefore all the weights are real, the stratification is due to the presence of weights of different lengths, while in the other cases it is due to the presence of complex weights.

Real closed fields and models of Peano arithmetic

2012 ◽  
Vol 75 (1) ◽  
pp. 1-11 ◽  
Author(s):  
P. D'Aquino ◽  
J. F. Knight ◽  
S. Starchenko
Keyword(s):  
Peano Arithmetic ◽  
Integer Part ◽  
The Real ◽  
Real Closed Fields ◽  
Nonstandard Model ◽  
Ordered Field ◽  
Closed Fields ◽  
Real Closure ◽  
Ordered Ring ◽  
Recursively Saturated

Shepherdson [14] showed that for a discrete ordered ring I, I is a model of I Open iff I is an integer part of a real closed ordered field. In this paper, we consider integer parts satisfying PA. We show that if a real closed ordered field R has an integer part I that is a nonstandard model of PA (or even IΣ4), then R must be recursively saturated. In particular, the real closure of I, RC (I), is recursively saturated. We also show that if R is a countable recursively saturated real closed ordered field, then there is an integer part I such that R = RC(I) and I is a nonstandard model of PA.

Separately Nash and arc‐Nash functions over real closed fields

2020 ◽  
Author(s):  
Wojciech Kucharz ◽  
Krzysztof Kurdyka ◽  
Ali El‐Siblani
Keyword(s):  
Real Closed Fields ◽  
Closed Fields
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