From: Siu Lok Shun <lssiu@hkusua.hku.hk>
Subject: Fields whose Algebraic Closures are finite dimensional extensions
Date: 9 Jul 2000 03:35:54 GMT
Newsgroups: sci.math
Summary: [missing]

This is Artin-Schreier Theorem ( N. Jacobson, Basic Algebra II \S 11.7 pp.
654)

Let C be an algebraically closed field, R a proper subfield of finite
codimension C ([C:R]<\infty). Then R is real closed and C=R(\sqrt{-1}).

Hope this help.

-- 
==============================================================================

From: wcw@math.psu.edu (William C Waterhouse)
Subject: Re: Fields whose Algebraic Closures are finite dimensional extensions
Date: 11 Jul 2000 21:17:36 GMT
Newsgroups: sci.math

In article <8k6b2l$diu$1@slb7.atl.mindspring.net>, 
pollen@netcom.com (David Pollen) writes:
>   I am curious about the situation where one has a field (of 
> characteristic zero) for which a finite extension of it is algebraicly
> closed. ...

This is settled by a theorem of Artin and Schreier.

I wrote out a self-contained proof (assuming only basic Galois theory)
in the paper

   "When one equation solves them all"  

   American Mathematical Monthly 92 (1985), 270-273.


William C. Waterhouse
Penn State