Let $k$ be a field and $k[x_1,x_2,x_3,y_1,y_2,y_3]$ a polynomial ring in 6 variables over $k$. How to prove that the ideal $(x_1y_2-x_2y_1,x_2y_3-x_3y_2,x_3y_1-x_1y_3)$ is prime in $k[x_1,x_2,x_3,y_1,y_2,y_3]$ ?
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You may get some mileage out of this post: http://math.stackexchange.com/questions/95217/methods-to-check-if-an-ideal-of-a-polynomial-ring-is-prime-or-at-least-radical – Derek Allums Jul 19 '16 at 12:48
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1See http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2450856#p2450856 for why your ideal is the kernel of a ring homomorphism between two integral domains (take $n=1$ and $m=2$). – darij grinberg Jul 19 '16 at 12:57
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1Use this result. – user26857 Jul 19 '16 at 13:05
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1Geometrically, Blowing up an irreducible variety yields an irreducible variety. – MooS Jul 19 '16 at 13:15