Compute the radical of $\langle z-3x, z-\frac{3}{2}y\rangle$ in $\mathbb{C}[x,y,z]$

Question

I wish to compute the radical of the ideal $\langle z-3x, z-\frac{3}{2}y\rangle$ in $\mathbb{C}[x,y,z]$ (having worked with the Nullstellensatz Theorem to get here).

I presume the radical is the ideal itself, but do not know how to show it. I've tried to show the ideal is prime (not sure if it actually is prime but a prime ideal is always radical) and also tried applying the definition of the radical directly, having no luck.

Any tips would be great please!

Answer 1

It is indeed prime (hence radical), because the quotient is isomorphic to $\mathbb{C}[t]$ which is an integral domain.

To check your understanding, construct such an isomorphism.