What if $\operatorname{char}\mathbb{K}$ is not $0$ or if $\mathbb{K}$ is not algebraically closed? (Nullstellensatz) 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. 
 A: If the field has characteristic $0$ (or, more generally, is perfect) then you can use Galois theory: the maximal ideals correspond to orbits of points in $\overline{K}^n$ by ${\rm Gal}(\overline{K}/K)$. Given a point $P=(a_1,\dots,a_n)$ in $\overline{K}^n$, evaluation of polynomials in $K[x_1,\dots,x_n]$ at $P$ is a $K$-alg. hom from polynomials to $\overline{K}$ whose kernel is a maximal ideal of $K[x_1,\dots,x_n]$. All maximal ideals arise in this way, and two points $P$ and $Q$ give rise to the same maximal ideal iff $Q = \sigma(P)$ for some $\sigma$ in ${\rm Gal}(\overline{K}/K)$.
A: 1) Characteristic $0$ is irrelevant: if $K$ is an algebraically closed field  of any  characteristic the maximal ideals of $K[x_1,\ldots,x_n]$ are the ideals $I_a=(x_1-a_1,...,x_n-a_n)$ consisting of the polynomials $f\in K[x_1,\ldots,x_n]$ vanishing on $a=(a_1,\ldots,a_n)\in K^n$.
2) For an arbitrary, not necessarily algebraically closed,  field $K$ the maximal ideals of  $K[x_1,\ldots,x_n]$ correspond to the closed points of affine space $\mathbb A^n_K$.
A rough description of these points is as follows:
Choose an algebraic closure $K^a$ of $K$. There is a canonical morphism of schemes $\mathbb A^n_{K^a} \to \mathbb A^n_K$, dual to the inclusion  $K[x_1,\ldots,x_n]\to K^a[x_1,\ldots,x_n]$.  The closed points of   $\mathbb A^n_K$ are the images of the  closed points of $\mathbb A^n_{K^a}$.
 Here are a few facts surrounding/interpreting  that geometric description :
a) A prime  ideal $\mathfrak p\subset K[x_1,\ldots,x_n]$  is maximal $\iff $ the extension $K\subset Frac(A/\mathfrak p)$ is finite.
b) An ideal $I\subset K[x_1,\ldots,x_n]$  is maximal $\iff $ there exists $a\in (K^a)^n$ such that $I$ is the zero set $$I_a= \lbrace f\in K[x_1,\ldots,x_n]\mid f(a)=0\rbrace $$   c) Given $a,b \in  (K^a)^n$ , the corresponding maximal ideals $I_a, I_b$ are equal $\iff$ there exists a $K$-automorphism $s\in Aut(K^a/K)$ such that $s(a_i)=b_i$. (Note that $K^a/K$ is not assumed Galois.)  
Geometrically b)  says that $\mathbb A^n_{K^a} \to \mathbb A^n_K$ is surjective and c) describes the fibers of that morphism as the orbits of $Aut(K^a/K)$ acting on $(K^a)^n$
