Find all maximal ideals of $\mathbb{R}[x,y]/(x^2+y^2)$

Question

How to find all maximal ideals of $\mathbb{R}[x,y]/(x^2+y^2)$?

I have no idea because $\mathbb{R}$ is not algebraically closed. Is there any good way?

Answer 1

Any ideal $I$ of $R=\mathbb{R}\left[x,y\right]/\left\langle x^2+y^2\right\rangle$ is of the form $I=I_0/\left\langle x^2+y^2\right\rangle$ for some ideal $\left\langle x^2+y^2\right\rangle\subseteq I_0\vartriangleleft\mathbb{R}\left[x,y\right]$. $I$ is maximal in $R$ if and only if $I_0$ is maximal in $\mathbb{R}\left[x,y\right]$. So your question is equivalent to finding all maximal ideals in $\mathbb{R}\left[x,y\right]$ which contain $\left\langle x^2+y^2\right\rangle$. See here for the classification of maximal ideals of $\mathbb{R}\left[x,y\right]$.