Some theories associated with algebraically closed fields

1980 ◽  
Vol 45 (2) ◽  
pp. 359-362 ◽  
Author(s):  
Chris Ash ◽  
John Rosenthal
Keyword(s):  
Characteristic Zero ◽  
Algebraic Independence ◽  
Transcendence Degree ◽  
Rational Field ◽  
Partial Differentiation ◽  
Decidable Theory ◽  
Closed Fields ◽  
Algebraically Closed Fields

We consider the effect on decidability of adding, to the decidable theory of algebraically closed fields of characteristic zero, relation symbols for algebraic independence or function symbols for differentiation. Our results show that the corresponding theories are usually undeeidable.Let k and K be algebraically closed fields of characteristic zero. Let K be an extension of k of transcendence degree n over k. Since k has characteristic 0, we may assume that the rational field, Q, is a subfield of k.Let Indn be the n-ary relation on K which holds for exactly those n-tuples from K which are algebraically independent over k.Let x1, …, xn be a transcendence base for K over k. For i = 1, 2, …, n, let Di: K → K be the partial differentiation function with respect to xi and this base.Let KnInd = (K, +, ·, Indn), n ≤ 1 and let KnDiff = (K, +, ·, D1, …, Dn), n ≤ 1 where K has transcendence degree n over k.We show that the theories of these structures are independent of k when k has infinite transcendence degree over Q, that KnDiff has undeeidable theory for n ≤ 1 and that KnInd has undeeidable theory for n ≤ 2. The theory of K1Ind is decidable.

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).

scholarly journals 𝔸1-contractibility of affine modifications

2019 ◽  
Vol 30 (14) ◽  
pp. 1950069
Author(s):  
Adrien Dubouloz ◽  
Sabrina Pauli ◽  
Paul Arne Østvær
Keyword(s):  
Characteristic Zero ◽  
Infinite Family ◽  
Total Space ◽  
Fiber Bundles ◽  
Higher Dimensional ◽  
Closed Fields ◽  
Smooth Affine ◽  
Algebraically Closed Fields

We introduce Koras–Russell fiber bundles over algebraically closed fields of characteristic zero. After a single suspension, this exhibits an infinite family of smooth affine [Formula: see text]-contractible [Formula: see text]-folds. Moreover, we give examples of stably [Formula: see text]-contractible smooth affine [Formula: see text]-folds containing a Brieskorn–Pham surface, and a family of smooth affine [Formula: see text]-folds with a higher-dimensional [Formula: see text]-contractible total space.

Recursively presentable prime models

1974 ◽  
Vol 39 (2) ◽  
pp. 305-309 ◽  
Author(s):  
Leo Harrington
Keyword(s):  
Decision Procedure ◽  
Positive Answer ◽  
Prime Model ◽  
Property A ◽  
Categorical Theory ◽  
Decidable Theory ◽  
Free Variables ◽  
Closed Fields ◽  
Countable Theory ◽  
Algebraically Closed Fields

It is well known that a decidable theory possesses a recursively presentable model. If a decidable theory also possesses a prime model, it is natural to ask if the prime model has a recursive presentation. This has been answered affirmatively for algebraically closed fields [5], and for real closed fields, Hensel fields and other fields [3]. This paper gives a positive answer for the theory of differentially closed fields, and for any decidable ℵ1-categorical theory.The language of a theory T is denoted by L(T). All languages will be presumed countable. An x-type of T is a set of formulas with free variables x, which is consistent with T and which is maximal in this property. A formula with free variables x is complete if there is exactly one x-type containing it. A type is principal if it contains a complete formula. A countable model of T is prime if it realizes only principal types. Vaught has shown that a complete countable theory can have at most one prime model up to isomorphism.If T is a decidable theory, then the decision procedure for T equips L(T) with an effective counting. Thus the formulas of L(T) correspond to integers. The integer a formula φ(x) corresponds to is generally called the Gödel number of φ(x) and is denoted by ⌜φ(x)⌝. The usual recursion theoretic notions defined on the set of integers can be transferred to L(T). In particular a type Γ is recursive with index e if {⌜φ⌝.; φ ∈ Γ} is a recursive set of integers with index e.

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.

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.

scholarly journals Perfect pseudo-algebraically closed fields are algebraically bounded

2004 ◽  
Vol 271 (2) ◽  
pp. 627-637 ◽  
Author(s):  
Zoé Chatzidakis ◽  
Ehud Hrushovski
Keyword(s):  
Closed Fields ◽  
Algebraically Closed Fields
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