I am hoping that someone can help me to better understand what it means for a polynomial map of affine variables to be finite. I have the following definition:
A polynomial map of algebraic sets $X\to Y$ is finite if the ring homomorphism k[Y]$\to $k[X] is an integral extension.
While I do understand this definition I am struggling to apply the results to anything but the most basic examples. For example I believe that the polynomial map $\mathbb{A}^2\to\mathbb{A}^2$ given by $(x,y)\mapsto(x^2,y^2)$ is a finite map because the map of coordinate rings $f:k[u,v]\to k[x,y]$ given by $p(u,v)\mapsto p(x^2,y^2)$ is an extension $k[x^2,y^2]\subset k[x,y]$ with integral generators. Therefore the map is finite.
However I am struggling with more complicated examples. For the map $\mathbb{A}\to X$, $t\mapsto (t^2,t^3)$ where $X=V(y^2-x^3)\subset \mathbb{A}^2$ for example, I do not know how to go about finding an integral extension that will show if it is finite. Is there another definition or theorem that I need to show if it is finite? (Of course it may be infinite which would explain why I can't find an integral extension, however I am also struggling with another example - to show if the map $V(yx^2+y^2)\to \mathbb{A}^1$, $(x,y)\mapsto y$ is finite- which leads me to believe that it may be finite and that I am just lacking somewhere in my understanding.)
Any help would be appreciated!