Let $\mathbf{R}$ be the field of real numbers. What are the generators of the maximal ideals of the polynomial ring $\mathbf{R}[x\_1, ... , x\_n]$? If instead of $\mathbf{R}$ one considers the field $\mathbf{C}$ of complex numbers, then Hilbert's Nullstellensatz implies that each maximal ideal $\mathfrak{m}$ of $\mathbf{C}[x\_1, ... , x\_n]$ is generated by $n$ generators of the form $x\_i-a\_i$. 



Maximal ideals in a polynomial ring over the real numbers.
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 $$(x\_1^2+a\_1, x\_2-a\_2, \ldots, x\_n-a\_n), \quad a\_i \in \mathbf{R}, \quad a\_1 >0.$$


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](http://www.mat.ucm.es/~arrondo/commalg.pdf), in particular Example 0.6 and Corollary 5.14.


