examples of integrally closed extensions

Example. [5] is not integrally closedMathworldPlanetmath, for u=1+52[5] is integral over [5] since u2-u-1=0, but u[5].

Example. R=[2,3] is not integrally closed. Note that (6+2)/2R, but that


and so (6+2)/2 is integral over since it satisfies the polynomialPlanetmathPlanetmath (z2-2)2-3=0.

Example. 𝒪K is integrally closed when [K:]<. For if uK is integral over 𝒪K, then 𝒪K𝒪K[u] are all integral extensions, so u is integral over , so u𝒪K by definition. In fact, 𝒪K can be defined as the integral closureMathworldPlanetmath of in K.

Example. [x,y]/(y2-x3). This is a domain because y2-x3 is irreduciblePlanetmathPlanetmath hence a prime idealMathworldPlanetmathPlanetmath. But this quotient ringMathworldPlanetmath is not integrally closed. To see this, parameterize [x,y][t] by

x t2
y t3

The kernel of this map is (y2-x3), and its image is [t2,t3]. Hence


and the field of fractionsMathworldPlanetmath of the latter ring is obviously (t). Now, t is integral over [t2,t3] (z2-t2 is its polynomial), but is not in [t2,t3]. t corresponds to yx in the original ring [x,y]/(y2-x3), which is thus not integrally closed (the minimal polynomial of yx is z2-x since (yx)2-x=y2x2-x=x3x2-x=0). The failure of integral closure in this coordinate ring is due to a codimension 1 singularity of y2-x3 at 0.

Example. A=[x,y,z]/(z2-xy) is integrally closed. For again, parameterize A[u,v] by

x u2
y v2
z uv

The kernel of this map is z2-xy and its image is B=[u2,v2,uv]. Claim B is integrally closed. We prove this by showing that the integral closure of [x,y] in (x,y,xy) is [x,y,xy]. Choose r+sxy(x,y,xy),r,s(x,y) such that r+sxy is integral over [x,y]. Then r-sxy is also integral over [x,y], so their sum is. Hence 2r is integral over [x,y]. But [x,y] is a UFD, hence integrally closed, so 2r[x,y] and thus r[x,y]. Similarly, sxy is integral over [x,y], hence s2xy[x,y],s(x,y). Clearly, then, s can have no denominator, so s[x,y]. Hence r+sxy[x,y,xy].

Title examples of integrally closed extensions
Canonical name ExamplesOfIntegrallyClosedExtensions
Date of creation 2013-03-22 17:01:32
Last modified on 2013-03-22 17:01:32
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 9
Author rm50 (10146)
Entry type Example
Classification msc 13B22
Classification msc 11R04