So let $\mathbf{F}\_q$ be a finite field with $q$ elements where $q=p^m$, 


$p$ a prime number and $m\in\mathbf{Z}\_{\geq 1}$. Let $f(x,y)\in \mathbf{F}\_q[x,y]$


be a smooth non-constant polynomial and let $A:=\mathbf{F}\_q[x,y]/(f)$. 


Q: Does there exists an integer $N\_0$ (which depends on $A$) such that for $N\geq N\_0$ one 


may always find a prime ideal $P\subseteq A$ such that $A/P\simeq\mathbf{F}\_{q^N}$? 



About points on affine curves defined over finite fields
The answer is "yes", given what we know about the number of points on the corresponding projective curve (or more accurately a smooth model of it, in case of singularities). Since points at infinity and singular points can be bounded terms of the degree of *f*, they don't affect the asymptotics here. So the question comes down to the existence of points with a given field of definition, of size a large power of *q*. The calculation that the main term gives the right answer (>0) can be read off the case of the affine line, where it is certainly true.


