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Given a field $\mathbb{K}$ which is algebraically closed and of characteristic 0, we can say exactly what the maximal ideals of $\mathbb{K}[x_1,\dots,x_n]$ are and they correspond to points in $\mathbb{K}^n$ (thanks to the Nullstellensatz).

Can I say anything about the maximal ideals of $\mathbb{K}$ if char$\mathbb{K}\neq0$ or if it is not algebraically closed?

Thanks.

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    The nullstellensatz does not require characteristic zero. Generalizations can be found here: http://en.wikipedia.org/wiki/Hilbert%27s_Nullstellensatz .2012-05-01
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    Even Hilbert's Nullstellensatz says something far more interesting than "maximal ideals = points": it says a polynomial function $f$ vanishes on the zero locus of an ideal $I$ of $k[x_1, \ldots, x_n]$ if and only if a power of $f$ is contained in $I$. This really and honestly does require the field to be algebraically closed.2012-05-01

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