Characterising maximal ideals in $\mathbb F_p[x]$, $\mathbb Z[x]$ I'm interested in characterising maximal ideals in $\mathbb F_p[x,y]$. More precisely, my problem is:

Find all possible cardinalities for fields of the kind $A/I$, where $A=\mathbb Z[x,y]/(x^2+y^2)$ and $I$ is a maximal ideal in $A$ (knowing that these cardinalities are all finite).

It is easy to obtain all fields of cardinality $p$, $p$ prime. What can we say about $p^n$?
And, more in general, what can we say about maximal ideals in $\mathbb F_p[x]$? (We can't use Hilbert's Nullstellensatz since $\mathbb F_p$ is not algebraically closed.)
Thank you in advance.
 A: Another easy case is the case $q=2^n$. Any finite field $\mathbf F_q$ is easily seen to be isomorphic to a quotient of the ring $A$ if $q=2^n$.
If $A/I$ is a finite field, other than the ones you found already and the ones above, then there is a surjecive ring morphism $f\colon A\rightarrow \mathbf F_q$, where $q=\#(A/I)$, $q=p^n$ with $p$ an odd prime and $n>1$. Let $\bar x$ and $\bar y$ be the images of $x$ and $y$ in $\mathbf F_q$. Since $f$ is surjective, $\mathbf F_q=\mathbf F_p(\bar x,\bar y)$. In particular, $\bar x$ and $\bar y$ cannot both be zero. Since $x^2+y^2=0$ in $A$, one has $\bar x^2+\bar y^2=0$ in $\mathbf F_q$. Hence $\bar x$ and $\bar y$ are both nonzero. Then, $(\bar x/\bar y)^2=-1$ in $\mathbf F_q$, i.e., $\mathbf F_q$ has a square root of $-1$. It follows that $q\equiv 1\pmod 4$.
Conversely, if $q\equiv1\pmod4$, let $i\in\mathbf F_q$ be a square root of $-1$, and let $\bar x$ be a generator of the field extension $\mathbf F_q/\mathbf F_p$. Let $\bar y=i\bar x$. Let $f\colon A\rightarrow \mathbf F_q$ be the morhism defined by $f(x)=\bar x$ and $f(y)=\bar y$. Then $f$ is surjective. It follows that $\mathbf F_q$ is isomorphic to a quotient $A/I$ of the ring $A$.
