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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On finding fields from their algebraic closure geometries

Author: Kitty L. Holland
Journal: Proc. Amer. Math. Soc. 116 (1992), 1135-1142
MSC: Primary 03C60; Secondary 12L12
MathSciNet review: 1100654
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Abstract: It is shown that if $ {F_1}$ and $ {F_2}$ are algebraically closed fields of nonzero characteristic $ p$ and $ {F_1}$ is not isomorphic to a subfield of $ {F_2}$, then $ {F_1}$ does not embed in the skew field of quotients $ {O_{{F_2}}}$ of the ring of morphisms of the additive group of $ {F_2}$. From this fact and results of Evans and Hrushovski, it is deduced that the algebraic closure geometries $ G({K_1}/{F_1})$ and $ G({K_2}/{F_2})$ are isomorphic if and only if $ {K_1}:{F_1} \simeq {K_2}:{F_2}$. It is further proved that if $ {F_0}$ is the prime algebraically closed field of characteristic $ p$ and $ F$ has positive transcendence degree over $ {F_0}$, then $ {O_F}$ and $ {O_{{F_0}}}$ are not elementarily equivalent.

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Keywords: Algebraically closed field, algebraic group, projective plane
Article copyright: © Copyright 1992 American Mathematical Society

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