TITLE: How to show a ring is normal or not, and how to show the normalisation of the ring
QUESTION [8 upvotes]: I am confused about how to show whether a ring is normal or not. For example, consider the $k$-algebra $k[x,y] /\langle x^2 - y^3 \rangle$, which is a domain. How do I show it is not normal?  Are there any standard techniques? I know that I want to show it is not integrally closed in its fraction field, but I can't work out how to do this, i.e. find some $z$ in the fraction field, integral over the ring but is not in the ring.
Also how do I work out what the normalisation actually is? Sorry I have no working to offer because I am completely stumped.

REPLY [6 votes]: There is an easy way to tell where the ring $k[x,y] / f(x,y)$ is not regular: it is precisely at the ideal
$$ \langle f(x,y), \partial_x f(x,y), \partial_y f(x,y) \rangle$$
This remains true in more variables and more equations -- take all first-derivatives of all equations. The geometric interpretation is that this ideal describes the singular set of your system of equations.
Since your ring is one-dimensional, being non-regular is synonymous with being non-normal.
Applying the test and assuming we aren't in characteristic 2 or 3, we get the ideal
$$ \langle x^2 - y^3, 2x, 3y^2 \rangle = \langle x, y^2 \rangle$$
so the ring is not normal. The normalization will involve adjoining fractions $u/v$ of things in this ideal. Alas, I haven't gone through this procedure often enough to be able to do it systematically, so I can't offer advice on that.