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.