On predicates in algebraically closed fields

1954 ◽  
Vol 19 (2) ◽  
pp. 103-114 ◽  
Author(s):  
Abraham Robinson
Keyword(s):  
Functional Calculus ◽  
General Structure ◽  
Curve Surface ◽  
Plane Curve ◽  
Coordinate Space ◽  
Closed Field ◽  
Algebraic Field ◽  
Closed Fields ◽  
Image Position ◽  
Algebraically Closed Fields

Many properties of curves, surfaces, or other varieties in Algebraic Geometry can be formulated in the lower functional calculus as predicates of the coefficients of the polynomial or polynomials which define the variety (curve, surface) in question. For example, the property of a plane curve of order n to possess exactly m double points, or the property to be of genus p — where m and p are specified integers — can be formulated in this way. Similarly many statements on the relation between two or more varieties, e.g., concerning the number and type of their intersection points, can be expressed in the lower functional calculus. It is usual to study the properties of a variety in an algebraically closed field. Accordingly, it is of considerable interest to investigate the general structure of the class of predicates mentioned above in relation to algebraically closed fields. The following result will be proved in the present paper.Main Theorem. Let F be a commutative algebraic field of arbitrary characteristic, and let F′ = F[x1, …, xn] be the ring of polynomials of n variables with coefficients in F. With every predicate Q{x1, …, xn) which is formulated in the lower functional calculus in terms of the relations of equality, addition, and multiplication and (possibly) in terms of some of the elements of F, there can be associated an ascending chain of ideals in F′,such thatfor every extension F* of F which is algebraically closed. In this formula, V0, …, V2k+ 1 are the varieties of the ideals J1, …, J2k+i in the coordinate space Sn: (x1, … xn) over F*, and VQ is the set of points of Sn which satisfy Q.

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.

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).

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 Parameter Spaces for Algebraic Equivalence

2017 ◽  
Vol 2019 (6) ◽  
pp. 1863-1893 ◽  
Author(s):  
Jeffrey D Achter ◽  
Sebastian Casalaina-Martin ◽  
Charles Vial
Keyword(s):  
Abelian Varieties ◽  
Closed Field ◽  
Parameter Spaces ◽  
Algebraically Closed Field ◽  
Arbitrary Base ◽  
Closed Fields ◽  
Algebraic Equivalence ◽  
Integral Scheme ◽  
The Difference ◽  
Algebraically Closed Fields

Abstract A cycle is algebraically trivial if it can be exhibited as the difference of two fibers in a family of cycles parameterized by a smooth integral scheme. Over an algebraically closed field, it is a result of Weil that it suffices to consider families of cycles parameterized by curves, or by abelian varieties. In this article, we extend these results to arbitrary base fields. The strengthening of these results turns out to be a key step in our work elsewhere extending Murre’s results on algebraic representatives for varieties over algebraically closed fields to arbitrary perfect fields.

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.

Definability in reducts of algebraically closed fields

1988 ◽  
Vol 53 (1) ◽  
pp. 188-199
Author(s):  
Gary A. Martin
Keyword(s):  
Rational Function ◽  
Equivalence Class ◽  
Positive Characteristic ◽  
Algebraic Function ◽  
Equivalence Classes ◽  
Closed Field ◽  
Additional Hypothesis ◽  
Closed Fields ◽  
Power Set ◽  
Algebraically Closed Fields

Let K be an algebraically closed field and let L be its canonical language; that is, L consists of all relations on K which are definable from addition, multiplication, and parameters from K. Two sublanguages L1 and L2 of L are definably equivalent if each relation in L1 can be defined by an L2-formula with parameters in K, and vice versa. The equivalence classes of sublanguages of L form a quotient lattice of the power set of L about which very little is known. We will not distinguish between a sublanguage and its equivalence class.Let Lm denote the language of multiplication alone, and let La denote the language of addition alone. Let f ∈ K [X, Y] and consider the algebraic function defined by f (x, y) = 0 for x, y ∈ K. Let Lf denote the language consisting of the relation defined by f. The possibilities for Lm ∨ Lf are examined in §2, and the possibilities for La ∨ Lf are examined in §3. In fact the only comprehensive results known are under the additional hypothesis that f actually defines a rational function (i.e., when f is linear in one of the variables), and in positive characteristic, only expansions of addition by polynomials (i.e., when f is linear and monic in one of the variables) are understood. It is hoped that these hypotheses will turn out to be unnecessary, so that reasonable generalizations of the theorems described below to algebraic functions will be true. The conjecture is that L covers Lm and that the only languages between La and L are expansions of La by scalar multiplications.

scholarly journals RIGID ORBITS AND SHEETS IN REDUCTIVE LIE ALGEBRAS OVER FIELDS OF PRIME CHARACTERISTIC

2016 ◽  
Vol 17 (3) ◽  
pp. 583-613 ◽  
Author(s):  
Alexander Premet ◽  
David I. Stewart
Keyword(s):  
Lie Algebras ◽  
Closed Field ◽  
Nilpotent Orbits ◽  
Good Characteristic ◽  
Related Condition ◽  
Exceptional Lie Algebras ◽  
Closed Fields ◽  
Simply Connected ◽  
Algebraically Closed Fields

Let$G$be a simple simply connected algebraic group over an algebraically closed field$k$of characteristic$p>0$with$\mathfrak{g}=\text{Lie}(G)$. We discuss various properties of nilpotent orbits in$\mathfrak{g}$, which have previously only been considered over$\mathbb{C}$. Using computational methods, we extend to positive characteristic various calculations of de Graaf with nilpotent orbits in exceptional Lie algebras. In particular, we classify those orbits which are reachable as well as those which satisfy a certain related condition due to Panyushev, and determine the codimension of the derived subalgebra$[\mathfrak{g}_{e},\mathfrak{g}_{e}]$in the centraliser$\mathfrak{g}_{e}$of any nilpotent element$e\in \mathfrak{g}$. Some of these calculations are used to show that the list of rigid nilpotent orbits in$\mathfrak{g}$, the classification of sheets of$\mathfrak{g}$and the distribution of the nilpotent orbits amongst them are independent of good characteristic, remaining the same as in the characteristic zero case. We also give a comprehensive account of the theory of sheets in reductive Lie algebras over algebraically closed fields of good characteristic.

Model theory of strictly upper triangular matrix rings

1980 ◽  
Vol 45 (3) ◽  
pp. 455-463 ◽  
Author(s):  
William H. Wheeler
Keyword(s):  
Model Theory ◽  
Triangular Matrix ◽  
Closed Field ◽  
Matrix Rings ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Closed Fields ◽  
Upper Triangular Matrix ◽  
Algebraically Closed Fields ◽  
Model Completion

Two questions on rings of strictly upper triangular matrices arising from B. Rose's work [5] are answered in this paper. An n × n matrix (αi, j) is strictly upper triangular if αi, j = 0 whenever i ≥ j. The ring of strictly upper triangular n × n matrices with entries from a field F will be denoted by Sn(F). Throughout this paper n will be an integer greater than 2. B. Rose [5] has shown that the complete theory of Sn(F) for an algebraically closed field F is ℵ1categorical. The first main result of this paper is that the rings Sn(F) and Sn(K) for fields F and K are isomorphic or elementarily equivalent if and only if F and K are isomorphic or elementarily equivalent, respectively (Corollary 1.6 and Theorem 2.2). This result shortens the proof of B. Rose's categoricity theorem [5, Theorem 7] by avoiding the co-stability considerations; furthermore, this result yields a proof of the converse of this categoricity theorem. The second main result is that the theory of rings of strictly upper triangular n × n matrices over algebraically closed fields is the model-completion of the theory of rings of strictly upper triangular n × n matrices over arbitrary fields (Theorem 2.5). This answers affirmatively the two conjectures at the end of [5].

EXPANSIONS OF ALGEBRAICALLY CLOSED FIELDS II: FUNCTIONS OF SEVERAL VARIABLES

2003 ◽  
Vol 03 (01) ◽  
pp. 1-35 ◽  
Author(s):  
YA'ACOV PETERZIL ◽  
SERGEI STARCHENKO
Keyword(s):  
Rational Function ◽  
Meromorphic Functions ◽  
Closed Field ◽  
Several Variables ◽  
Closed Fields ◽  
Weierstrass Preparation ◽  
Definable Functions ◽  
Algebraically Closed Fields ◽  
Compact Complex Manifolds ◽  
Definable Function

Let ℛ be an o-minimal expansion of a real closed field R. We continue here the investigation we began in [11] of differentiability with respect to the algebraically closed field [Formula: see text]. We develop the basic theory of such K-differentiability for definable functions of several variables, proving theorems on removable singularities as well as analogues of the Weierstrass preparation and division theorems for definable functions. We consider also definably meromorphic functions and prove that every definable function which is meromorphic on Kn is necessarily a rational function. We finally discuss definable analogues of complex analytic manifolds, with possible connections to the model theoretic work on compact complex manifolds, and present two examples of "nonstandard manifolds" in our setting.

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