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Let $k$ be a field and $k[x_1,x_2,x_3,x_4]$ a polynomial ring in four variables over $k$. How can we show that the ideal $(x_3^3-x_2^2x_4, x_4^3-x_1^2x_3, x_3x_4-x_1x_2, x_2x_4^2-x_1x_3^2)$ is prime? I always find this kind of problem is not easy to me, is there a criterion for this kind of problem?

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    $\begingroup$ Isn't this the kernel of the map $k[x_1,x_2,x_3,x_4] \to k[s^4, t^4, st^3, s^3t]$? If so, I believe it was answered before here. In general, showing something is prime is a hard question. $\endgroup$ – Youngsu Dec 7 '14 at 9:55
  • $\begingroup$ @nick If don't want to use a computer program, then I suggest you to have a look at this answer. $\endgroup$ – user26857 Dec 7 '14 at 10:51

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