0

I would like to know if the quotient ring $R=\Bbb Q[X,Y]/(X-Y^2,Y-X^2)$ can be described in an easier way, i.e. is it isomorphic to something known?

If I denote by $x$ the class of $X$ and by $y$ the class of $Y$, then $x=y^2,y=x^2$, so that $x(x^3-1)=0$ so if the ring is a domain, then I have a third root of unity. Then maybe $R$ is related to $\Bbb Q(\zeta_3)=\Bbb Q(i\sqrt 3)$.

My question is a bit similar to this one. I don't know algebraic geometry, but maybe this is related to this topic.

Thank you!

Community
  • 1
Alphonse
  • 6,342
  • 1
  • 19
  • 48

1 Answers1

4

This ring is just isomorphic to $\mathbb Q[X]/(X-X^4) \cong \mathbb Q[X]/(X) \times \mathbb Q[X]/(X-1) \times \mathbb Q[X]/(X^2+X+1) \cong \mathbb Q \times \mathbb Q \times \mathbb Q(\zeta_3)$

MooS
  • 31,390