Is $\mathbb{Q} [X,Y]/[x^{20},y^{20}]$ a local ring?

Question

Is $\mathbb{Q}[X,Y]/[x^{20},y^{20}]$ is a local ring?

My approach is to look for a maximal ideal, but got stuck how to find a single maximal ideal. Any feedback would be appreciated.

Answer 1

Well, $\Bbb Q[X,Y]/\langle X^{20},Y^{20}\rangle$ is a local ring. The ideal $\langle X,Y\rangle$ is maximal in $\Bbb Q[X,Y]$, and as $\langle X,Y\rangle^{40}\subseteq\langle X^{20},Y^{20}\rangle \subseteq\langle X,Y\rangle$ the radical of $\langle X^{20},Y^{20}\rangle$ is $\langle X,Y\rangle$. This means that the image of $\langle X,Y\rangle$ in $\Bbb Q[X,Y]/\langle X^{20},Y^{20}\rangle$ is the unique maximal ideal.

Answer 2

One characterization of local rings is that they have a proper ideal $m$ whose complement consists entirely of units.

In your case, the ideal $(X,Y)$ of $\mathbb Q[X,Y]$ is a (proper) ideal containing $(X^{20},Y^{20})$ and so its image in the quotient is a (proper) ideal. Any element of the complement clearly has to be represented by some polynomial $f(X,Y)$ with nonzero constant term. Then you can write it as $$f(X,Y) = c + Xg(X,Y) + Yh(X,Y)$$ In the quotient, $c$ is still a unit and both $X$ and $Y$ are nilpotent. As the sum of a unit and a nilpotent is a unit, we see that $f(X,Y)$ is a unit and so by the fact mentioned above this ring is local.