$\mathbb{Q}[x,y]/(f(x,y))$ is not a field for any irreducible $f(x,y) \in \mathbb{Q}[x,y]$.

Question

I need to show that $\mathbb{Q}[x,y]/(f(x,y))$ is not a field for any irreducible $f(x,y) \in \mathbb{Q}[x,y]$.

My approach was to find a bigger proper ideal containing $f(x,y)$ but i am unable to proceed.

Answer 1

Write $f(x,y)=a_n(x)y^n+\cdots+a_0(x)$, with $a_i(x)\in \mathbb{Q}[x]$, $a_n(x)\neq 0$.

If $n=0$, the case is easy. We deal with $n\geq 1$. Find an integer $m$ such that $a_n(m)a_n(m+1)\neq 0$. Then $f(x,y)\in (f(x,y),x-m)$, also $f\in (f,x-m-1)$. Show that ideals $(f(x,y),x-m)$, $(f,x-m-1)$ are not $(1)$. And if $(f(x,y))$ were maximal, then $(f)=(f(x,y),x-m)=(f,x-m-1)$, then $1\in (f)$. But as shown before the ideal $(f(x,y),x-m)$ is not $(1)$.