MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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: I looked at 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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.