Consider $R=\mathbb Q[x,y]/(x^2+y^2+1)$.
Q1: Are there any field extensions of $\mathbb Q$ besides degree 2 extensions by quotienting out maximal ideals? I could not find any polynomial generates degree 3 extension or higher. I considered $I=(xy-1)+(x^2+y^2+1)$, and it looks like I got a degree 4 extension from this as $R/I$ is a finite dimensional $\mathbb Q$-vector space. Correct me if I am wrong.
Q2: Is this a Dedekind ring?