TITLE: Methods to check if an ideal of a polynomial ring is prime or at least radical
QUESTION [26 upvotes]: I am looking for methods to check whether a given ideal in $K[x_0,\dots,x_n]$ is prime. I mean something you can effectively use in some concrete non-trivial example.
To be more explicit, I am working in $K[x_0,\dots,x_3]$ ($K$ algebraically closed) and I have got these four polynomials:
$f_1=x_0x_3-x_1x_2,$
$f_2=x_0^2x_2-x_1^3,$
$f_3=x_1x_3^2-x_2^3,$
$f_4=x_0x_2^2-x_1^2x_3$.
Let then $J=(f_1,f_2,f_3)$ and $I=(f_1,f_2,f_3,f_4)$. I would like to prove that $I=\operatorname{rad}(J)$. One can easily check that $f_4^2 \in J$, so we have $I\subset \operatorname{rad}(J)$. On the other hand we have $J\subset I\subset \operatorname{rad}(J)\subset \operatorname{rad}(I)$, and so $\operatorname{rad}(I)=\operatorname{rad}(J)$.
And here is my problem: it would suffice to show that $I$ is prime or at least radical to have what we want, but I find $I$ complicated enough to check this directly frome the definition and first properties. So I am asking if anyone knows some method which can easily be applied to this ideal (which I know to be prime thanks to a computer algebra system..but I need a proof). I notice that $I$ is a homogeneous ideal, maybe this can help, I don't know if there is any ad hoc method for homogeneous ideals.

REPLY [4 votes]: By using the same technique I've used here one can prove that $K[x_0,\dots,x_3]/I$ is isomorphic to $K[s^4,s^3t,st^3,t^4]$. In particular, this shows that $I$ is a prime ideal.