From the Hilbert's Nullstellensatz, we know that for any algebraic closed field $K$ and any prime ideal $p$ of $K[X_1,X_2,\cdots,X_n]$, the intersection of all the maximal ideals containing $p$ is $p$. Does this conclusion still hold for a general field $K$?
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6$\begingroup$ Yes, see en.wikipedia.org/wiki/Jacobson_ring $\endgroup$– YCorCommented Oct 25, 2015 at 12:32
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