Calculate the radical

Question

How to calculate the radical of the ideal $\langle zw^3,xyzw,x^5y^6\rangle$ in $k[x,y,z,w]$?

I know that the radical is the intersection of all minimal prime ideals. Then I considered a minimal prime ideal and find the possibilities $\langle y,w\rangle$, $\langle y,z\rangle$, $\langle x,w\rangle$ and $\langle x,z\rangle$.

I can not develop more than that. I accept suggestions. Thx.

Answer 1

You are on the right path. $\langle y,w\rangle\cap\langle y,z\rangle=\langle y,wz\rangle$ and $\langle x,w\rangle\cap\langle x,z\rangle=\langle x,wz\rangle$, so the answer you are looking for is $\langle y,wz\rangle\cap\langle x, wz\rangle=\langle yx,wz\rangle$.