William H. Wheeler. Model-complete theories of pseudo-algebraically closed fields. Annals of mathematical logic, vol. 17 (1979), pp. 205–226.

1987 ◽  
Vol 52 (4) ◽  
pp. 1055-1056
Author(s):  
Alexander Prestel
Keyword(s):  
Mathematical Logic ◽  
Closed Fields ◽  
Complete Theories ◽  
Algebraically Closed Fields

Model-complete theories of formally real fields and formally p-adic fields

1983 ◽  
Vol 48 (4) ◽  
pp. 1130-1139
Author(s):  
William H. Wheeler
Keyword(s):  
Galois Groups ◽  
General Situation ◽  
Complete Model ◽  
Yield Model ◽  
Valued Fields ◽  
Subsequent Investigation ◽  
Ordered Fields ◽  
Closed Fields ◽  
Complete Theories ◽  
Algebraically Closed Fields

The complete, model-complete theories of pseudo-algebraically closed fields were characterized completely in [11]. That work constituted the first step towards determining all the model-complete theories of fields in the usual language of fields. In this paper the second step is taken. Namely, the methods of [11] are extended to characterize the complete, model-complete theories of pseudo-real closed fields and pseudo-p-adically closed fields.In order to unify the treatment of these two types of fields, the relevant properties of real closed ordered fields and p-adically closed valued fields are abstracted. The subsequent investigation of model-complete theories of fields is based entirely on these properties. The properties were selected in order to solve three problems: (1) finding universal theories with the joint embedding property, (2) finding first order conditions in the usual language of fields which are necessary and sufficient for a polynomial over a field to have a zero in a formally real or formally p-adic extension of that field, and (3) finding subgroups of Galois groups whose fixed fields are formally real or formally p-adic.This paper is related to, and uses in §1 but not in the other sections, parts of K. McKenna's work [8] on model-complete theories of ordered fields and p-valued fields. However, the results herein are not direct consequences of his work, both because these results apply to a more general situation and because they use a different formal language. Concerning the latter point, in some instances, such as real closed ordered fields and p-adically closed valued fields, model-complete theories in expanded languages do yield model-complete theories of ordinary fields other than theories of pseudo-algebraically closed fields. However, in other cases, such as differentially closed fields, this is not so.

scholarly journals Coding complete theories in Galois groups

2008 ◽  
Vol 73 (2) ◽  
pp. 474-491
Author(s):  
James Gray
Keyword(s):  
Galois Groups ◽  
Vietoris Topology ◽  
Closed Fields ◽  
Prime Fields ◽  
Complete Theories ◽  
Pseudofinite Fields ◽  
Algebraically Closed Fields

AbstractIn this paper, I will give a new characterisation of the spaces of complete theories of pseudofinite fields and of algebraically closed fields with a generic automorphism (ACFA) in terms of the Vietoris topology on absolute Galois groups of prime fields.

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.

scholarly journals Adelic descent theory

2017 ◽  
Vol 153 (8) ◽  
pp. 1706-1746
Author(s):  
Michael Groechenig
Keyword(s):  
Vector Bundles ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Reductive Algebraic Group ◽  
Perfect Complexes ◽  
Descent Theory ◽  
Closed Fields ◽  
Ring Of Continuous Functions ◽  
Reconstruction Theorem ◽  
Algebraically Closed Fields

A result of André Weil allows one to describe rank $n$ vector bundles on a smooth complete algebraic curve up to isomorphism via a double quotient of the set $\text{GL}_{n}(\mathbb{A})$ of regular matrices over the ring of adèles (over algebraically closed fields, this result is also known to extend to $G$-torsors for a reductive algebraic group $G$). In the present paper we develop analogous adelic descriptions for vector and principal bundles on arbitrary Noetherian schemes, by proving an adelic descent theorem for perfect complexes. We show that for Beilinson’s co-simplicial ring of adèles $\mathbb{A}_{X}^{\bullet }$, we have an equivalence $\mathsf{Perf}(X)\simeq |\mathsf{Perf}(\mathbb{A}_{X}^{\bullet })|$ between perfect complexes on $X$ and cartesian perfect complexes for $\mathbb{A}_{X}^{\bullet }$. Using the Tannakian formalism for symmetric monoidal $\infty$-categories, we conclude that a Noetherian scheme can be reconstructed from the co-simplicial ring of adèles. We view this statement as a scheme-theoretic analogue of Gelfand–Naimark’s reconstruction theorem for locally compact topological spaces from their ring of continuous functions. Several results for categories of perfect complexes over (a strong form of) flasque sheaves of algebras are established, which might be of independent interest.

scholarly journals Intersections of algebraically closed fields

1986 ◽  
Vol 30 (2) ◽  
pp. 103-119 ◽  
Author(s):  
C.J. Ash ◽  
John W. Rosenthal
Keyword(s):  
Closed Fields ◽  
Algebraically Closed Fields
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