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There are two kind of maximal ideals in $\mathbf{R}[x_1, \ldots, x_n]$: the ideals corresponding to real points of $\mathbf{A}^n_{\mathbb{R}}$, i.e. of the form $$(x_1-a_1, \ldots, x_n-a_n), \quad a_i \in \mathbf{R}$$ and the ideals corresponding to pairs of complex-conjugated points, that after a real change of coordinates can be put in the form $$(x_1^2+a_1, x_2-a_2, \ldots, x_n-a_n), \quad a_i \in \mathbf{R}, \quad a_1 >0 \textrm{ not a square}.$$0.$$This follows from the generalized weak Nullstellensatz, saying that if \mathbf{K} is any field and \mathfrak{m} \subset \mathbf{K}[x_1, \ldots, x_n] is a maximal ideal, then the field \mathbf{K}[x_1, \ldots, x_n]/ \mathfrak{m} is a finite extension of \mathbf{K}. In particular, when \mathbf{K}=\mathbf{R} this extension must have degree at most 2. For a reference, see Arrondo's notes A geometric introduction to commutative algebra, in particular Example 0.6 and Corollary 5.14. 3 added 23 characters in body There are two kind of maximal ideals in \mathbf{R}[x_1, \ldots, x_n]: the ideals corresponding to real points of \mathbf{A}^n_{\mathbb{R}}, i.e. of the form$$(x_1-a_1, \ldots, x_n-a_n), \quad a_i \in \mathbf{R}$$and the ideals corresponding to pairs of complex-conjugated points, that after a real change of coordinates can be put in the form$$(x_1^2+a_1, x_2-a_2, \ldots, x_n-a_n), \quad a_i \in \mathbf{R}, \quad a_1 >0.$$0 \textrm{ not a square}.$$

This follows from the generalized weak Nullstellensatz, saying that if $\mathbf{K}$ is any field and $\mathfrak{m} \subset \mathbf{K}[x_1, \ldots, x_n]$ is a maximal ideal, then the field $\mathbf{K}[x_1, \ldots, x_n]/ \mathfrak{m}$ is a finite extension of $\mathbf{K}$. In particular, when $\mathbf{K}=\mathbf{R}$ this extension must have degree at most $2$.

For a reference, see Arrondo's notes A geometric introduction to commutative algebra, in particular Example 0.6 and Corollary 5.14.

2 added 24 characters in body

There are two kind of maximal ideals in $\mathbf{R}[x_1, \ldots, x_n]$: the ideals corresponding to real points of $\mathbf{A}^n_{\mathbb{R}}$, i.e. of the form $$(x_1-a_1, \ldots, x_n-a_n), \quad a_i \in \mathbf{R}$$ and the ideals corresponding to pairs of complex-conjugated points, that after a real change of coordinates can be put in the form $$(x_1^2+a, x_2(x_1^2+a_1, x_2-a_2, \ldots, x_n)x_n-a_n), \quad a a_i \in \mathbf{R}_+.$$mathbf{R}, \quad a_1 >0.

This follows from the generalized weak Nullstellensatz, saying that if $\mathbf{K}$ is any field and $\mathfrak{m} \subset \mathbf{K}[x_1, \ldots, x_n]$ is a maximal ideal, then the field $\mathbf{K}[x_1, \ldots, x_n]/ \mathfrak{m}$ is a finite extension of $\mathbf{K}$. In particular, when $\mathbf{K}=\mathbf{R}$ this extension must have degree at most $2$.

For a reference, see Arrondo's notes A geometric introduction to commutative algebra, in particular Example 0.6 and Corollary 5.14.

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