# Artin Schreier Theorem for Rings

This has been in my mind for quite some time. Looking at Artin Schreier Theorem for fields:

If L is a field and K its algebraic closure and if 1< [K:L] < infinity then L=K[i] and L is a real closed field (Thus L has characteristic 0. Here i is just the square root of -1).

I was wondering if a "generalized" Artin Schreier exist or if someone could refer to me to some paper that attempts this. There is a concept of real closedness and "algebraic closedness" of reduced commutative rings, but I doubt that the statement would hold.

So one has the following conjecture:

If L is a reduced commutative ring and K is its total integral closure (this is an equivalent notion of algebraic closure if K and L were fields) and if 1<[K:L]< infinity (here I mean that K is a finite L-module that is not the same as L) then L is real (thus its characteristic is 0.. and one can add that L is real closed in the sense of reduced commutative rings).

Can one easily show this, even at least prove that L has characteristic 0?

-
Do you mean the <b> absolute </b> integral closure ? –  Hailong Dao Jan 2 '10 at 4:34
I am not familiar with the term absolute integral closure. Could you provide a reference or define it for me? Definition of total integral closure (some also call it algebraic closure) for reduced commutative ring can be seen here: tinyurl.com/yah3zds. I looked at tinyurl.com/y8lpal4 and there it defines the absolute integral closure. I believe I can prove that TIC and AIC are equal notions given the definition I saw for AIC. –  Jose Capco Jan 2 '10 at 6:09
@Dao : I just browsed your website.. Since your advisor was Mel Hochster, I should have at least mentioned his paper on total integrally closed ring : Totally integrally closed rings and extremal spaces, Pacific Journal of Math., Vol. 32, No. 3, 1970. Robert Raphael, Edgar Enochs and Borho were a few other people in the 70's who worked on this. –  Jose Capco Jan 2 '10 at 6:19
The AIC definition was what I had in mind. I still do not understand the definition of TIC in the planetmath link you gave. It says: if R is domain, then TIC is the algebraic closure of the function field. Then shouldn't what you want follow from Artin-Schreier? Also, if that's the definition of TIC, then how can it be equivalent to AIC, which is the integral closure in some algebraic closure of the function field? –  Hailong Dao Jan 2 '10 at 7:51
Yes you are right, the planetmath remark was wrong (I wrote it). It should have been the integral closure of the domain in the algebraic closure of its quotient field. I just corrected the planetmath entry. –  Jose Capco Jan 2 '10 at 8:09