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In the one variable case ($k[x]$ where $k$ is a field), we know that a polynomial is irreducible iff the ideal it generates is maximal. Now, reviewing the proof, it is clear that this proof doesn't carry over to the multivariable case, but are there any conditions on when a polynomial in $k[x_1,x_2,...,x_n]$ generates a maximal ideal? (We can assume k is a algebraically closed field).

david h
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    Every maximal ideal $M$ then is of this form, i..e, $M=(x_1-a_1,...,x_n-a_n)$. – Dietrich Burde Jan 02 '23 at 11:48
  • Is there a way to show that maximal ideal generated by those n polynomials cannot be generated by one polynomial or cannot be generated by fewer than n polynomials? – david h Jan 02 '23 at 11:52
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    Yes, there is. You can take this post, for example. Actually, there is also a nice post here:"No maximal ideal of $k[x_1, \dots, k_n]$ is principal for $n > 1$" So $(x_1-a_1,\ldots ,x_n-a_n)=(f)$ is of course impossible, for $n\ge 2$. Does this answer your question? – Dietrich Burde Jan 02 '23 at 12:11

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