How to compute the height of $(xy-z^2, yz-x^2, zx-y^2)$ in $k[x,y,z]$?

Question

Firstly, how to prove that $\mathfrak{p} = (xy-z^2, yz-x^2, zx-y^2)$ is a prime ideal of $k[x,y,z]$? Then, how could I verify that $\operatorname{height}\mathfrak{p} = 2$?

Answer 1

Who gave you this problem? The ideal is not prime.

We clearly have $\mathfrak p \subset (x-y,y-z)$. If the LHS would be prime, both sides would be primes of height $2$, hence equality would hold. But the LHS is generated by homogenous elements of degree $2$, hence the LHS does not contain any linear polynomials.

Conclusion: $\mathfrak p$ is not prime.