The classification of small weakly minimal sets. III: Modules

1988 ◽  
Vol 53 (3) ◽  
pp. 975-979 ◽  
Author(s):  
Steven Buechler
Keyword(s):  
Finite Field ◽  
Algebraic Closure ◽  
Morley Rank ◽  
Minimal Sets

AbstractTheorem A. Let M be a left R-module such that Th(M) is small and weakly minimal, but does not have Morley rank 1. Let A = acl(∅) ⋂ M and I = {r ∈ R: rM ⊂ A}. Notice that I is an ideal.(i) F = R/Iis a finite field.(ii) Suppose that a, b0,…,bn, ∈ M and . Then there are s, ri ∈ R, i ≤ n, such that sa + Σi≤nribi ∈ A and s ∉ I.It follows from Theorem A that algebraic closure in M is modular. Using this and results in [B1] and [B2], we obtainTheorem B. Let M be as in Theorem A. Then Vaught's conjecture holds for Th(M).

scholarly journals On algebraic closure in pseudofinite fields

2012 ◽  
Vol 77 (4) ◽  
pp. 1057-1066 ◽  
Author(s):  
Özlem Beyarslan ◽  
Ehud Hrushovski
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Algebraic Closure ◽  
Roots Of Unity ◽  
Prime Field ◽  
Almost All ◽  
Pseudofinite Fields

AbstractWe study the automorphism group of the algebraic closure of a substructureAof a pseudo-finite fieldF. We show that the behavior of this group, even whenAis large, depends essentially on the roots of unity inF. For almost all completions of the theory of pseudofinite fields, we show that overA, algebraic closure agrees with definable closure, as soon asAcontains the relative algebraic closure of the prime field.

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.

Pseudoprojective strongly minimal sets are locally projective

1991 ◽  
Vol 56 (4) ◽  
pp. 1184-1194 ◽  
Author(s):  
Steven Buechler
Keyword(s):  
Predicate Symbol ◽  
Infinite Dimension ◽  
Elementary Extension ◽  
Morley Rank ◽  
Minimal Set ◽  
Unary Predicate ◽  
Minimal Sets

AbstractLet D be a strongly minimal set in the language L, and D′ ⊃ D an elementary extension with infinite dimension over D. Add to L a unary predicate symbol D and let T′ be the theory of the structure (D′, D), where D interprets the predicate D. It is known that T′ is ω-stable. We proveTheorem A. If D is not locally modular, then T′ has Morley rank ω.We say that a strongly minimal set D is pseudoprojective if it is nontrivial and there is a k < ω such that, for all a, b ∈ D and closed X ⊂ D, a ∈ cl(Xb) ⇒ there is a Y ⊂ X with a ∈ cl(Yb) and ∣Y∣ ≤ k. Using Theorem A, we proveTheorem B. If a strongly minimal set D is pseudoprojective, then D is locally projective.The following result of Hrushovski's (proved in §4) plays a part in the proof of Theorem B.Theorem C. Suppose that D is strongly minimal, and there is some proper elementary extension D1 of D such that the theory of the pair (D1, D) is ω1-categorical. Then D is locally modular.

scholarly journals Classification of Elliptic Cubic Curves Over The Finite Field of Order Nineteen

2016 ◽  
Vol 13 (4) ◽  
pp. 846-852
Author(s):  
Baghdad Science Journal
Keyword(s):  
Finite Field ◽  
Elliptic Curve ◽  
Maximum Size ◽  
Cubic Curves ◽  
Projective Equivalence

Plane cubics curves may be classified up to isomorphism or projective equivalence. In this paper, the inequivalent elliptic cubic curves which are non-singular plane cubic curves have been classified projectively over the finite field of order nineteen, and determined if they are complete or incomplete as arcs of degree three. Also, the maximum size of a complete elliptic curve that can be constructed from each incomplete elliptic curve are given.

scholarly journals GENERALIZED GELFAND–GRAEV REPRESENTATIONS IN SMALL CHARACTERISTICS

2016 ◽  
Vol 224 (1) ◽  
pp. 93-167 ◽  
Author(s):  
JAY TAYLOR
Keyword(s):  
Finite Field ◽  
Wave Front ◽  
Irreducible Character ◽  
Prime Order ◽  
Algebraic Closure ◽  
Characteristic Functions ◽  
Rational Structure ◽  
Reductive Algebraic Group ◽  
Wave Front Set ◽  
Frobenius Endomorphism

Let $\mathbf{G}$ be a connected reductive algebraic group over an algebraic closure $\overline{\mathbb{F}_{p}}$ of the finite field of prime order $p$ and let $F:\mathbf{G}\rightarrow \mathbf{G}$ be a Frobenius endomorphism with $G=\mathbf{G}^{F}$ the corresponding $\mathbb{F}_{q}$-rational structure. One of the strongest links we have between the representation theory of $G$ and the geometry of the unipotent conjugacy classes of $\mathbf{G}$ is a formula, due to Lusztig (Adv. Math. 94(2) (1992), 139–179), which decomposes Kawanaka’s Generalized Gelfand–Graev Representations (GGGRs) in terms of characteristic functions of intersection cohomology complexes defined on the closure of a unipotent class. Unfortunately, the formula given in Lusztig (Adv. Math. 94(2) (1992), 139–179) is only valid under the assumption that $p$ is large enough. In this article, we show that Lusztig’s formula for GGGRs holds under the much milder assumption that $p$ is an acceptable prime for $\mathbf{G}$ ($p$ very good is sufficient but not necessary). As an application we show that every irreducible character of $G$, respectively, character sheaf of $\mathbf{G}$, has a unique wave front set, respectively, unipotent support, whenever $p$ is good for $\mathbf{G}$.

The canonical topology on dp-minimal fields

2018 ◽  
Vol 18 (02) ◽  
pp. 1850007 ◽  
Author(s):  
Will Johnson
Keyword(s):  
Algebraic Closure ◽  
University Of California ◽  
Symbolic Logic ◽  
Valued Fields ◽  
Stepping Stone ◽  
Definable Type ◽  
Definable Sets ◽  
Type V

We construct a nontrivial definable type V field topology on any dp-minimal field [Formula: see text] that is not strongly minimal, and prove that definable subsets of [Formula: see text] have small boundary. Using this topology and its properties, we show that in any dp-minimal field [Formula: see text], dp-rank of definable sets varies definably in families, dp-rank of complete types is characterized in terms of algebraic closure, and [Formula: see text] is finite for all [Formula: see text]. Additionally, by combining the existence of the topology with results of Jahnke, Simon and Walsberg [Dp-minimal valued fields, J. Symbolic Logic 82(1) (2017) 151–165], it follows that dp-minimal fields that are neither algebraically closed nor real closed admit nontrivial definable Henselian valuations. These results are a key stepping stone toward the classification of dp-minimal fields in [Fun with fields, Ph.D. thesis, University of California, Berkeley (2016)].

scholarly journals Character sheaves and generalized Springer correspondence

2003 ◽  
Vol 170 ◽  
pp. 47-72 ◽  
Author(s):  
Anne-Marie Aubert
Keyword(s):  
Finite Field ◽  
Algebraic Group ◽  
Algebraic Closure ◽  
Reductive Algebraic Group ◽  
Good For ◽  
Character Sheaves ◽  
Springer Correspondence

AbstractLetGbe a connected reductive algebraic group over an algebraic closure of a finite field of characteristicp. Under the assumption thatpis good forG, we prove that for each character sheafAonGwhich has nonzero restriction to the unipotent variety ofG, there exists a unipotent classCAcanonically attached toA, such thatAhas non-zero restriction onCA, and any unipotent classCinGon whichAhas non-zero restriction has dimension strictly smaller than that ofCA.

scholarly journals A GENERATION THEOREM FOR GROUPS OF FINITE MORLEY RANK

2008 ◽  
Vol 08 (02) ◽  
pp. 163-195 ◽  
Author(s):  
JEFFREY BURDGES ◽  
GREGORY CHERLIN
Keyword(s):  
Morley Rank ◽  
Generation Theorem ◽  
Finite Morley Rank

We deal with two forms of the "uniqueness cases" in the classification of large simple K*-groups of finite Morley rank of odd type, where large means the 2-rank m2(G) is at least three. This substantially extends results known for even larger groups having Prüfer 2-rank at least three, so as to cover the two groups PSp 4 and G 2. With an eye towards more distant developments, we carry out this analysis for L*-groups, a context which is substantially broader than the K* setting.

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 TOWARDS A CLASSIFICATION OF FINITE FIELD THEORIES

1994 ◽  
Vol 09 (27) ◽  
pp. 2555-2567
Author(s):  
PETER GRANDITS
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Finite Field ◽  
Field Theories ◽  
Quantum Field ◽  
Particle Content ◽  
Finiteness Conditions ◽  
Gauge Groups ◽  
Yukawa Couplings

We consider the finiteness conditions on the Yukawa couplings of a general quantum field theory for gauge groups SU (n)(n>6) and a rather general particle content. It is shown that in the class of theories considered (149 different particle contents), only two models are able to fulfill the finiteness conditions. Only one of these is supersymmetric. For the nonsupersymmetric one the appropriate Yukawa couplings are constructed explicitly.

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