# Number of maximal ideals of $F_q[x_1,…,x_n]$

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?

• Use LaTeX please: math.meta.stackexchange.com/questions/5020 – Michael Rozenberg Jun 28 at 14:06
• There are $\frac{1}{k}\sum_{d | k} \mu(d) q^{k/d}$ maximal ideals such that $F_q[x_1]/m = F_{q^k}$. For $n \ge 1$ it is worth looking at the ($\log$ of the ) zeta function of $\Bbb{A}^n_{F_q}$ – reuns Jun 28 at 14:17
• Is there another way that you can think of? I'm not familiar with the zeta function @reuns – Ariel Jun 28 at 17:15

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.
• So, let me see if I understood you right: Let $q=p^n$.Followed by the first isomorphism theorem,we know that there is an isomorphism $\alpha:F_{p^n}[x_1,…,x_n]/ker(\alpha)→F_{p^k}$ where k divides n(because we want the right side to be a field).So,it follows that $ker(α)$ is a maximal ideal.Hence,the number of maximal ideals is equal to the number of surjective homomorphisms(is it just the number of surjective fuctions from A to B in general?)minus the number of surjective homomorphisms which have the same maximal ideal($=p^{k−1}∗p^{k−1}∗p^{k−2}∗...∗1$).@reuns – Ariel Jun 29 at 13:22
• @Ariel $f_n(k) = q^{nk}$, $g_n(k) = \sum_{d | k} \mu(d) f_n(k/d)$ and the number of maximal ideals is $\frac{1}{k} g_n(k)$ (because $x\to a, x \to b$ have the same kernel iff $a = (b_1^{q^r},\ldots,b_n^{q^r})$ for some $r$) – reuns Jun 29 at 14:52