The Failure of Integral Closure
What Singularities look like to Algebra
Recall that a ring is integrally closed if it is integral over its field of fractions e.g. any element of Frac which is a root of a monic polynomial with coefficients in is contained in For example, is integrally closed.
Consider . The polynomial is a root of the (monic) polynomial and so is integral over However, And for those who like pictures: the geometric incarnation of this fact is that the plane curve has a singularity.