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Show that the ideal $I = \langle x^{2m}+y^{2m}-1\rangle$ is a maximal ideal in the polynomial ring $\mathbb{R}[x,y]$

How can I prove that
$$I = \langle x^{2m}+y^{2m}-1\rangle \ (\,m \in \mathbb{N}\, )$$
is a maximal ideal in the polynomial ring $\mathbb{R}[x,y]$?
I think that there’s a straightforward way, without thinking about the elements that are not in $I$.

A:

Suppose that $I\subset J$ is an ideal such that $J
eq I$.
By the definition of ideals, there exist $f\in I$, and $g\in J$ such that
$fg\in I\cap J=\{0\}$, which is a contradiction.
Hence, $I$ is maximal.

A:

This is just a duplicate of [1].
Proof of Note: Because every subring of the field of reals is a quotient ring of the polynomial ring $\mathbb{R}[x,y]$, the ideal $\langle x^{2m}+y^{2m}-1\rangle$ is maximal in $\mathbb{R}[x,y]$.

[1]

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