Abraham Robinson. On predicates in algebraically closed fields. The journal of symbolic logic, vol. 19 (1954), pp. 103–114.

1960 ◽  
Vol 25 (2) ◽  
pp. 169-170
Author(s):  
W. Peremans
Keyword(s):  
Symbolic Logic ◽  
Closed Fields ◽  
Algebraically Closed 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

Pseudo Algebraically Closed Fields

1986 ◽  
pp. 129-140
Author(s):  
Michael D. Fried ◽  
Moshe Jarden
Keyword(s):  
Closed Fields ◽  
Algebraically Closed Fields
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