scholarly journals Motives for perfect PAC fields with pro-cyclic Galois group

2008 ◽  
Vol 73 (3) ◽  
pp. 1036-1050
Author(s):  
Immanuel Halupczok
Keyword(s):  
Galois Group ◽  
Finite Fields ◽  
Grothendieck Ring ◽  
Additional Information ◽  
Chow Motives ◽  
Closed Fields ◽  
Cohomological Invariant ◽  
Definable Sets ◽  
Grothendieck Rings ◽  
Algebraically Closed Fields

AbstractDenef and Loeser denned a map from the Grothendieck ring of sets definable in pseudo-finite fields to the Grothendieck ring of Chow motives, thus enabling to apply any cohomological invariant to these sets. We generalize this to perfect, pseudo algebraically closed fields with pro-cyclic Galois group.In addition, we define some maps between different Grothendieck rings of definable sets which provide additional information, not contained in the associated motive. In particular we infer that the map of Denef-Loeser is not injective.

Properties of forking in ω-free pseudo-algebraically closed fields

2002 ◽  
Vol 67 (3) ◽  
pp. 957-996 ◽  
Author(s):  
Zoé Chatzidakis
Keyword(s):  
Galois Group ◽  
Finite Fields ◽  
Order Theory ◽  
Absolute Galois Group ◽  
Prime Field ◽  
First Order ◽  
First Order Theory ◽  
Closed Fields ◽  
The Absolute ◽  
Algebraically Closed Fields

The study of pseudo-algebraically closed fields (henceforth called PAC) started with the work of J. Ax on finite and pseudo-finite fields [1]. He showed that the infinite models of the theory of finite fields are exactly the perfect PAC fields with absolute Galois group isomorphic to , and gave elementary invariants for their first order theory, thereby proving the decidability of the theory of finite fields. Ax's results were then extended to a larger class of PAC fields by M. Jarden and U. Kiehne [21], and Jarden [19]. The final word on theories of PAC fields was given by G. Cherlin, L. van den Dries and A. Macintyre [10], see also results by Ju. Ershov [13], [14]. Let K be a PAC field. Then the elementary theory of K is entirely determined by the following data:• The isomorphism type of the field of absolute numbers of K (the subfield of K of elements algebraic over the prime field).• The degree of imperfection of K.• The first-order theory, in a suitable ω-sorted language, of the inverse system of Galois groups al(L/K) where L runs over all finite Galois extensions of K.They also showed that the theory of PAC fields is undecidable, by showing that any graph can be encoded in the absolute Galois group of some PAC field. It turns out that the absolute Galois group controls much of the behaviour of the PAC fields. I will give below some examples illustrating this phenomenon.

Combinatorics with Definable Sets: Euler Characteristics and Grothendieck Rings

2000 ◽  
Vol 6 (3) ◽  
pp. 311-330 ◽  
Author(s):  
Jan Krajíček ◽  
Thomas Scanlon
Keyword(s):  
Euler Characteristic ◽  
Order Structure ◽  
Euler Characteristics ◽  
Grothendieck Ring ◽  
Open Problems ◽  
Partially Ordered ◽  
Counting Functions ◽  
Finite Structures ◽  
Definable Sets ◽  
Grothendieck Rings

AbstractWe recall the notions of weak and strong Euler characteristics on a first order structure and make explicit the notion of a Grothendieck ring of a structure. We define partially ordered Euler characteristic and Grothendieck ring and give a characterization of structures that have non-trivial partially ordered Grothendieck ring. We give a generalization of counting functions to locally finite structures, and use the construction to show that the Grothendieck ring of the complex numbers contains as a subring the ring of integer polynomials in continuum many variables. We prove the existence of a universal strong Euler characteristic on a structure. We investigate the dependence of the Grothendieck ring on the theory of the structure and give a few counter-examples. Finally, we relate some open problems and independence results in bounded arithmetic to properties of particular Grothendieck rings.

On splitting of the normalizer of a maximal torus in orthogonal groups

2016 ◽  
Vol 16 (09) ◽  
pp. 1750174 ◽  
Author(s):  
Alexey Galt
Keyword(s):  
Finite Fields ◽  
Maximal Torus ◽  
Positive Characteristic ◽  
Orthogonal Groups ◽  
Type Formula ◽  
Closed Fields ◽  
Algebraically Closed Fields

We answer the question of splitting of the normalizer of a maximal torus in orthogonal groups of type [Formula: see text] and [Formula: see text] over finite fields and algebraically closed fields of positive characteristic.

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.

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

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.

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