How can I verify that the ideal $(x^2-zw, z^2-yw, y^3-xw, w^3-xy^2z)$ is prime in $\mathbb Q[x,y,z,w]$? I want to show that the ideal $$(x^2-zw, z^2-yw, y^3-xw, w^3-xy^2z)$$ in the ring $\mathbb{Q}[x,y,z,w]$ is prime, how can I?

If you believe in computers, you can do this in Macaulay2 with the following commands.
i1 : R = QQ[x,y,z,w]

o1 = R

o1 : PolynomialRing

i2 : I = ideal(x^2-z*w, z^2-y*w, y^3-x*w, w^3-x*y^2*z)

             2         2         3             2     3
o2 = ideal (x  - z*w, z  - y*w, y  - x*w, - x*y z + w )

o2 : Ideal of R

i3 : isPrime I

o3 = true

Macaulay2 can be run online for free at http://habanero.math.cornell.edu:3690/. 
