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.

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