Number of maximal ideals of $F_q[x_1,...,x_n]$

Question

I am currently studying commutative algebra and came across the following question.

Let $F$ be a finite field with $q$ elements, let $A=F[x_1,...,x_n]$ and denote by $m$ a maximal ideal in $A$.

1. How many maximal ideals are in $A$ such that $A/m = F$ ?
2. How many maximal ideals are in $A$ such that $A/m = L$ , where $|L| = q^k$ ?
3. How many maximal ideals are in $A$ ?

I know that maximal ideals of $F[x_1,...,x_n]$, where $F$ is an algebraically closed field are of the form $(x-a_1,...,x-a_n)$, but how does a maximal ideal looks like in that kind of a situation?

Answer 1

Let $f_n(k)$ be the number of morphisms $\Bbb{F}_q[x_1,\ldots,x_n] \to \Bbb{F}_{q^k}$.

They are also morphisms $\Bbb{F}_q[x_1,\ldots,x_n] \to \Bbb{F}_{q^{dk}}$ for every $d$. Thus we can use inclusion exclusion to count the number $g_n(k)$ of them being surjective.

The kernel of such a morphism is a maximal ideal, how many morphisms have the same kernel ?