Failure of Integral Closure

The Failure of Integral Closure

What Singularities look like to Algebra

Recall that a ring $R$ is integrally closed if it is integral over its field of fractions e.g. any element of Frac$(R)$ which is a root of a monic polynomial with coefficients in $R$ is contained in $R.$ For example, $\mathbb{Z}$ is integrally closed.

Consider $\mathbb{C}[t^2,t^3]$. The polynomial $t = t^3/t^2 \in \mathbb{C}(t)$ is a root of the (monic) polynomial $x^2 - t^2$ and so is integral over $\mathbb{C}[t^2,t^3].$ However, $t \notin \mathbb{C}[t^2,t^3].$ And for those who like pictures: the geometric incarnation of this fact is  that the plane curve $Y^2 - X^3 = 0$ has a singularity.