I'm looking to find the dimension of the scheme Spec $k[w, x, y, z]/(wz−xy, y^{17}+z^{17})$ where k is an algebraically closed field.
So far I've found that Spec $k[w, x, y, z]/(wz−xy)$ has dimension 3 because $(wz-xy)$ is irreducible, and not a zero divisor in k and thus you can use Krull's Principal Ideal theorem to show that it has codimension 1, and thus the scheme has dimension 3.
I'm stuck on the second step though since I'm not sure how to prove $y^{17}+z^{17}$ is irreducible.
I'd appreciate any help or hints for this problem.