How to show an ideal isn't radical

Question

The ideal in question is $(wy-x^2,w^2z-x^3)$ in the ring $k[w,x,y,z]$ for algebraically closed $k$.

Answer 1

First, you need to find a simple polynomial that is zero in the zero-set $V(I)$ of the ideal $I$. If $(a,b,c,d)$ is in $V(I)$, then $abd=a^3=d^2c$, so either $d=0$ or $ab=cd$. But if $d=0$, then $a^2=0$, so $ab=cd$ again. Hence the polynomial $xy-zw$ is zero in all $V(I)$. Therefore $xy-zw$ is in the radical of $I$. (One can show also that $(xy-zw)^3$ is in $I$, but you don't need this.)

Now you only need to show that $xy-zw$ is not in $I$.