Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-1992-1100654-6
MathSciNet review: 1100654
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] C. J. Ash and John W. Rosenthal, Intersections of algebraically closed fields, Ann. Pure Appl. Logic 30 (1986), 103-119. MR 829314 (87i:03090)
  • [2] D. Evans and E. Hrushovski, Projective planes in algebraically closed fields, Proc. London Math. Soc. (3) 62 (1991), 1-24. MR 1078211 (92a:05031)
  • [3] B. Lindström, On $ p$-polynomial representations of projective geometries in algebraic combinatorial geometries, Math. Scand. 63 (1988), 36-42. MR 994968 (90d:51012)
  • [4] J. Silverman, The arithmetic of elliptic curves, Springer-Verlag, Berlin and New York, 1986. MR 817210 (87g:11070)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 03C60, 12L12

Retrieve articles in all journals with MSC: 03C60, 12L12


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1100654-6
Keywords: Algebraically closed field, algebraic group, projective plane
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society