One of the ways I find more useful to check if a given ideal $I$ of $K[X_1,\ldots,X_n]$ is prime, is to look at the quotient ring $K[X_1,\ldots,X_n]/I$.
If I'm able to show it is isomorphic to $K[f_1,\ldots,f_n]$ where $f_i$ are polynomials in several variables then I'm done.
For example, to check that $I=(x_1x_3-x_2^2,x_0x_2-x_1^2,x_0x_3-x_1x_2)$ is prime it is enough to show that the homomorphism $\varphi:D[x_0,x_1,x_2,x_3]\rightarrow K[u,v]$ that takes $x_0$ to $v^3$, $x_1$ to $uv^2$, $x_2$ to $u^2v$ and $x_3$ to $u^3$ has as kernel exactly $I$. And this can be easily done with Groebner Basis. So we have shown that the quotient ring is isomorphic to $K[v^3,v^2u,vu^2,u^3]$ which is clearly an integral domain.
So my question is if this can always be done, ie given a prime ideal $I$ of $K[x_1,\ldots,x_n]$ is it always possible to find an isomorfism between the quotient ring and $K[f_1,\ldots,f_n]$ where $f_i$ are polynomials in several variables.
If that is not possible, what do we have to ask to $I$ or to $V(I)$ so this is possible?
In case it can be done, is there any way to find these polynomials?