What is the field of fractions of $\mathbb{Q}[x,y]/(x^2+y^2)$?

abstract-algebra ring-theory commutative-algebra

What is the field of fractions of $\mathbb{Q}[x,y]/(x^2+y^2)$?

Remarks:

(1) I think it is clear that $\mathbb{Q}[x,y]/(x^2+y^2)$ is an integral domain; indeed, $x^2+y^2 \in \mathbb{Q}[x,y]$ is irreducible (by considerations of degrees) hence prime.

(2) The field of fractions of $\mathbb{Q}[x,y]/(x^2+y^2-1)$ is isomorphic to $\mathbb{Q}(t)$, see this question and also this question.


First do the substitution $x'=x/y$. Then the equation $x^2+y^2=0$ transforms to $x'^2+1=0$. Hence we are looking at the (isomorphic) field $\text{Frac}(\mathbb Q[x',y]/(x'^2+1))$.

This is just $\mathbb Q(i)(y)$.

Related

Classes, sets and Russell's paradox
Is indefinite integration largely a heuristic or it can be mechanical too?
Can there be a symbol for continuous product? [closed]
Span of a subset of a vector space is the smallest subspace containing that set
Why would you take the logarithmic derivative of a generating function?
Transitive action of a discrete group on a compact space
Integral $\int_{-\infty}^\infty \frac{\exp{(-x^2)}}{1+x^4}dx$
Expected time to fuse a $D_{10}$ dragon
Why does this Infinite Series have contradictory convergences?
Is the Diophantine equation $y(x^3-y)=z^2+2$ solvable?
Every subset of $\mathbb{R}$ with finite measure is the disjoint union of a finite number of measurable sets
Four men, hats and probability