Talk:Integrally closed

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Untitled[edit]

There is another definition of integral closure in algebraic number theory.

And in algebraic geometry/commutative algebra. I am adding that one.

The definition for ordered groups is unclear. What is written can be interpretted as: "for all a and b in G, and all natural n, (if a^n < b then a < 1 )"

I think what is intended is: "for all a and b in G, ( if for all natural n (a^n < b) then a < 1 )"

Someone should look into this.

relation to "Integrality"[edit]

The Integrally Closed page requests suggestions on whether or not it should be merged with "Integrality." The "Integrality" page has discussions to the algebraic geometry and other uses. The Integrally Closed page request for discussion points to an Integrality talk page. DeaconJohnFairfax (talk) 22:39, 30 June 2008 (UTC) 6/30/2008[reply]

Problem with explanation[edit]

I don't think this is right:

"That is, every monic polynomial with coefficients in R has all of its roots in R."

For example, the integers are integrally closed in their fraction field (the rationals) but not every monic polynomial with integer coefficients has all it's root in the integers (i.e. x^2+1). I think it should read something like:

"That is, every monic polynomial with coefficients in R has NO roots in S which are not in R."

--Paul Laroque (talk) 20:55, 26 April 2009 (UTC)[reply]