How to show that $\mathbb{C}[x_1,x_2,x_3, x_4]/(x_1x_2 - x_3x_4, x_1x_3 - x_2x_4, x_1x_4 - x_2x_3)$ is an integral domain?

Question

I am looking for a way to show that the ring $\mathbb{C}[x_1,x_2,x_3, x_4]/I$ where $$I = (x_1x_2 - x_3x_4, x_1x_3 - x_2x_4, x_1x_4 - x_2x_3)$$ is an integral domain. In other words I want to show $I$ is prime in $\mathbb{C}[x_1, x_2, x_3, x_4]$.

It would be great if someone could also give me a general way of doing these $x_ix_j - x_kx_l$ sort of problems as well.

Answer 1

This ideal is in fact not prime: notice that

$$ \begin{align*}(x_1^2 - x_4^2)(x_2^2 - x_3^2) &= x_1^2x_2^2 - x_1^2x_3^2 - x_4^2x_2^2 + x_3^2x_4^2 \\ &\equiv (x_1x_2)(x_3x_4) - (x_1x_3)(x_2x_4) - (x_4x_2)(x_1x_3) + (x_3x_4)(x_1x_2) \\ &= 0 \pmod I \end{align*}$$

If $I$ were prime, then one of $x_1 \pm x_4$, $x_2 \pm x_3 \in I$, but $I$ contains no polynomials of degree $1$, being generated by polynomials of degree $\ge 2$.