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