Differentially algebraic group chunks

1990 ◽  
Vol 55 (3) ◽  
pp. 1138-1142 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Algebraic Group ◽  
Characteristic Zero ◽  
Quantifier Elimination ◽  
Closed Field ◽  
Zariski Topology ◽  
First Order ◽  
Differential Field ◽  
Closed Fields ◽  
Differential Fields ◽  
Algebraically Closed Fields

We point out that a group first order definable in a differentially closed field K of characteristic 0 can be definably equipped with the structure of a differentially algebraic group over K. This is a translation into the framework of differentially closed fields of what is known for groups definable in algebraically closed fields (Weil's theorem).I restrict myself here to showing (Theorem 20) how one can find a large “differentially algebraic group chunk” inside a group defined in a differentially closed field. The rest of the translation (Theorem 21) follows routinely, as in [B].What is, perhaps, of interest is that the proof proceeds at a completely general (soft) model theoretic level, once Facts 1–4 below are known.Fact 1. The theory of differentially closed fields of characteristic 0 is complete and has quantifier elimination in the language of differential fields (+, ·,0,1, −1,d).Fact 2. Affine n-space over a differentially closed field is a Noetherian space when equipped with the differential Zariski topology.Fact 3. If K is a differentially closed field, k ⊆ K a differential field, and a and are in k, then a is in the definable closure of k ◡ iff a ∈ ‹› (where k ‹› denotes the differential field generated by k and).Fact 4. The theory of differentially closed fields of characteristic zero is totally transcendental (in particular, stable).

Uniqueness and characterization of prime models over sets for totally transcendental first-order theories

1972 ◽  
Vol 37 (1) ◽  
pp. 107-113 ◽  
Author(s):  
Saharon Shelah
Keyword(s):  
Simple Proof ◽  
Finite Sequence ◽  
The Other ◽  
Closed Field ◽  
Prime Model ◽  
First Order ◽  
Differential Field ◽  
Closed Fields ◽  
Transcendental Theory

If T is a complete first-order totally transcendental theory then over every T-structure A there is a prime model unique up to isomorphism over A. Moreover M is a prime model over A iff: (1) every finite sequence from M realizes an isolated type over A, and (2) there is no uncountable indiscernible set over A in M.The existence of prime models was proved by Morley [3] and their uniqueness for countable A by Vaught [9]. Sacks asked (see Chang and Keisler [1, question 25]) whether the prime model is unique. After proving this I heard Ressayre had proved that every two strictly prime models over any T-structure A are isomorphic, by a strikingly simple proof. From this followsIf T is totally transcendental, M a strictly prime model over A then every elementary permutation of A can be extended to an automorphism of M. (The existence of M follows by [3].)By our results this holds for any prime model. On the other hand Ressayre's result applies to more theories. For more information see [6, §0A]. A conclusion of our theorem is the uniqueness of the prime differentially closed field over a differential field. See Blum [8] for the total transcendency of the theory of differentially closed fields.We can note that the prime model M over A is minimal over A iff in M there is no indiscernible set over A (which is infinite).

Differential Polynomials

2017 ◽  
Author(s):  
Matthias Aschenbrenner ◽  
Lou van den Dries ◽  
Joris van der Hoeven
Keyword(s):  
Characteristic Zero ◽  
Natural Setting ◽  
Differential Polynomials ◽  
Differential Field ◽  
Closed Fields ◽  
Basic Facts ◽  
Differential Fields

This chapter deals with differential polynomials. It first presents some basic facts about differential fields that are of characteristic zero with one distinguished derivation, along with their extensions. It then considers various decompositions of differential polynomials in their natural setting, along with valued differential fields and the property of continuity of the derivation with respect to the valuation topology. It also discusses the gaussian extension of the valuation to the ring of differential polynomials and concludes with some basic results on simple differential rings and differentially closed fields. In contrast to the corresponding notions for fields, the chapter shows that differential fields always have proper d-algebraic extensions, and the differential closure of a differential field K is not always minimal over K.

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.

A note on valuation definable expansions of fields

1998 ◽  
Vol 63 (2) ◽  
pp. 739-743 ◽  
Author(s):  
Deirdre Haskell ◽  
Dugald Macpherson
Keyword(s):  
Valuation Ring ◽  
Quantifier Elimination ◽  
Real Closed Field ◽  
Closed Field ◽  
Real Closed Fields ◽  
Ordered Field ◽  
Valued Field ◽  
Closed Fields ◽  
Binary Predicate ◽  
Algebraically Closed Fields

In this note, we consider models of the theories of valued algebraically closed fields and convexly valued real closed fields, their reducts to the pure field or ordered field language respectively, and expansions of these by predicates which are definable in the valued field. We show that, in terms of definability, there is no structure properly between the pure (ordered) field and the valued field. Our results are analogous to several other definability results for reducts of algebraically closed and real closed fields; see [9], [10], [11] and [12]. Throughout this paper, definable will mean definable with parameters.Theorem A. Let ℱ = (F, +, ×, V) be a valued, algebraically closed field, where V denotes the valuation ring. Let A be a subset ofFndefinable in ℱv. Then either A is definable in ℱ = (F, +, ×) or V is definable in.Theorem B. Let ℛv = (R, <, +, ×, V) be a convexly valued real closed field, where V denotes the valuation ring. Let Abe a subset ofRndefinable in ℛv. Then either A is definable in ℛ = (R, <, +, ×) or V is definable in.The proofs of Theorems A and B are quite similar. Both ℱv and ℛv admit quantifier elimination if we adjoin a definable binary predicate Div (interpreted by Div(x, y) if and only if v(x) ≤ v(y)). This is proved in [14] (extending [13]) in the algebraically closed case, and in [4] in the real closed case. We show by direct combinatorial arguments that if the valuation is not definable then the expanded structure is strongly minimal or o-minimal respectively. Then we call on known results about strongly minimal and o-minimal fields to show that the expansion is not proper.

RAISING TO POWERS IN ALGEBRAICALLY CLOSED FIELDS

2003 ◽  
Vol 03 (02) ◽  
pp. 217-238 ◽  
Author(s):  
B. ZILBER
Keyword(s):  
Characteristic Zero ◽  
Order Theory ◽  
First Order ◽  
First Order Theory ◽  
Closed Fields ◽  
Algebraically Closed Fields

We study structures on the fields of characteristic zero obtained by introducing (multi-valued) operations of raising to power. Using Hrushovski–Fraisse construction we single out among the structures exponentially-algebraically closed once and prove, under certain Diophantine conjecture, that the first order theory of such structures is model complete and every its completion is superstable.

Superstable differential fields

1992 ◽  
Vol 57 (1) ◽  
pp. 97-108 ◽  
Author(s):  
A. Pillay ◽  
Ž. Sokolović
Keyword(s):  
Differential Forms ◽  
Quantifier Elimination ◽  
Normal Extension ◽  
Additional Structure ◽  
Unique Type ◽  
Generic Type ◽  
Differential Field ◽  
Closed Fields ◽  
Invariant Differential ◽  
Differential Fields

In this paper we study differential fields of characteristic 0 (with perhaps additional structure) whose theory is superstable. Our main result is that such a differential field has no proper strongly normal extensions in the sense of Kolchin [K1]. This is an approximation to the conjecture that a superstable differential field is differentially closed (although we believe the full conjecture to be false). Our result improves earlier work of Michaux [Mi] who proved that a (plain) differential field with quantifier elimination has no proper Picard-Vessiot extension. Our result is a generalisation of Michaux's, due to the fact that any plain differential field K with quantifier elimination is ω-stable. (Any quantifier free type over K defines a unique type over K in the sense of dc(k), the differential closure of K, and as we mention below the theory of differentially closed fields is ω-stable.) The proof of our main result depends on (i) Kolchin's theory [K3] which states that any strongly normal extension L of an algebraically closed differential field K is generated over K by an element η of some algebraic group G defined over CK, the constants of K, where η satisfies some specific differential equations over K related to invariant differential forms on G (η is “G-primitive” over K), and (ii) the fact that a superstable field has a unique generic type which is semiregular.

THE INDISCERNIBLE TOPOLOGY: A MOCK ZARISKI TOPOLOGY

2001 ◽  
Vol 01 (01) ◽  
pp. 99-124 ◽  
Author(s):  
MARKUS JUNKER ◽  
DANIEL LASCAR
Keyword(s):  
Order Structure ◽  
Zariski Topology ◽  
Topological Properties ◽  
First Order ◽  
Closed Fields ◽  
Structure Formula ◽  
Algebraically Closed Fields ◽  
Stable Structures

We associate with every first order structure [Formula: see text] a family of invariant, locally Noetherian topologies (one topology on each Mn). The structure is almost determined by the topologies, and properties of the structure are reflected by topological properties. We study these topologies in particular for stable structures. In nice cases, we get a behaviour similar to the Zariski topology in algebraically closed fields.

scholarly journals Nilpotent orbits of exceptional Lie algebras over algebraically closed fields of bad characteristic

1985 ◽  
Vol 38 (3) ◽  
pp. 330-350 ◽  
Author(s):  
D. F. Holt ◽  
N. Spaltenstein
Keyword(s):  
Finite Field ◽  
Lie Algebras ◽  
Closed Field ◽  
Nilpotent Orbits ◽  
Reductive Algebraic Group ◽  
Exceptional Lie Algebras ◽  
Closed Fields ◽  
Characteristic 3 ◽  
Algebraically Closed Fields

AbstractThe classification of the nilpotent orbits in the Lie algebra of a reductive algebraic group (over an algebraically closed field) is given in all the cases where it was not previously known (E7 and E8 in bad characteristic, F4 in characteristic 3). The paper exploits the tight relation with the corresponding situation over a finite field. A computer is used to study this case for suitable choices of the finite field.

scholarly journals Involutions on finite-dimensional algebras over real closed fields

2004 ◽  
Vol 77 (1) ◽  
pp. 123-128 ◽  
Author(s):  
W. D. Munn
Keyword(s):  
Characteristic Zero ◽  
Real Closed Field ◽  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Algebraically Closed Field ◽  
Real Closed Fields ◽  
Finite Dimensional ◽  
Closed Fields ◽  
Finite Dimensional Algebras

AbstractIt is shown that the following conditions on a finite-dimensional algebra A over a real closed field or an algebraically closed field of characteristic zero are equivalent: (i) A admits a special involution, in the sense of Easdown and Munn, (ii) A admits a proper involution, (iii) A is semisimple.

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