## 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.

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