suppose we have, in the ring $\mathbb{Z} [x,y,z,w,v]$, the following polynomials:
$f=xw-yz$,
$g=x^2z-y^3$,
$h=yw^2-z^3$,
$k=xz^2-y^2w$.
The question is to prove that $I=(f,g,h,k)$ is the radical ideal of $J=(f,g,h)$.
I tried several things to no avail: by imposing $x>y>z>w$ I tried to divide $k^2$ by $J$, but since I don't know whether $f,g,h$ is a Grobner base I stopped since having the residual being non zero wouldn't have said anything about it (also, the residual is not zero in this case). I then remembered that if $J$ is primary, the radical is the smallest prime who contains it. But checking that $J$ is primary seems like hell honestly. Since those are not simple polynomials, checking if any power of $k$ is contained in $J$ seems crazy. I could use some kind of software to find a Grobner base of $J$, but I'd like to do it by my own hand, and since my professor didn't explain how to find Grobner bases, I don't think it's the intended method to solve the exercise. Am I comitting some judgement errors here? How do I prove the thesis? Can anybody help? Thank you in advance.