What is the radical of this ideal in $\mathbb Q[x,y,z]$?

Question

What is the radical of the ideal $(x^5,y^3z^2,y^2xz^3,z^5,x^4y^2)$ in $\mathbb Q[x,y,z]$?

If we were working in $\mathbb C[x,y,z]$ then we could use the Nullstellensatz to show the radical is $(x,z)$. I am unsure how you would go about calculating the radical when we have $\mathbb Q$ instead of $\mathbb C$.

Answer 1

Certainly $(x,z)$ is contained in the radical of this ideal $I$ (we have $x^5 \in I$ and $z^5 \in I$). Conversely, $(x,z)$ is prime, hence radical, so must coincide with the radical of $I$.