On finding fields from their algebraic closure geometries

Holland, Kitty L.

doi:10.1090/S0002-9939-1992-1100654-6

On finding fields from their algebraic closure geometries
HTML articles powered by AMS MathViewer

by Kitty L. Holland

Proc. Amer. Math. Soc. 116 (1992), 1135-1142

DOI: https://doi.org/10.1090/S0002-9939-1992-1100654-6

PDF | Request permission

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

C. J. Ash and John W. Rosenthal, Intersections of algebraically closed fields, Ann. Pure Appl. Logic 30 (1986), no. 2, 103–119. MR 829314, DOI 10.1016/0168-0072(86)90001-1
David M. Evans and Ehud Hrushovski, Projective planes in algebraically closed fields, Proc. London Math. Soc. (3) 62 (1991), no. 1, 1–24. MR 1078211, DOI 10.1112/plms/s3-62.1.1
Bernt Lindström, On $p$-polynomial representations of projective geometries in algebraic combinatorial geometries, Math. Scand. 63 (1988), no. 1, 36–42. MR 994968, DOI 10.7146/math.scand.a-12223
Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8