# How to define affine/projective varieties over fields which are not algebraically closed?

In general, you come across affine and projective varieties as defined over algebraically closed fields, such as in Hartshorne's Algebraic Geometry, who defines

• an algebraic variety as an irreducible algebraic subset of $$\mathbb{A}^n$$, endowed with the induced topology.
• a projective variety as an irreducible algebraic subset of $$\mathbb{P}^n$$, endowed with the induced topology.

My questions:

Thank you!

• I suppose the problem is that in the non-algebraically closed field, there is no more a one-to-one correspondence between points in varieties and maximal ideals in $k$-algebras. Hilbert's Nullstellensatz tells us that in the alg. closed field, a point $(a,b,c)$ in a variety corresponds to the maximal ideal $(x-a,y-b,c-z)$. However, for example in $\mathbb{R}[x]$, this does not happen. To be specific, there is no point associated with the maximal ideal $(x^2+1)$.
– J1U
Oct 7, 2018 at 7:49
• And also, I am not quite sure about this, but since every field $k$ has an algebraic closure $\bar{k}$, so polynomials that generates the ideal in $k[x_1,\ldots,x_n]$ can be viewed as polynomials of $\bar{k}[x_1,\ldots,x_n]$, and if we know all about $\bar{k}[x_1,\ldots,x_n]$, then maybe we automatically get to know all about $k[x_1,\ldots,x_n]$. At least I think every algebraic set in $k^n$ can be viewed as the intersection of an algebraic set in $\bar{k}^n$ and $k^n \subset \bar{k}^n$.
– J1U
Oct 7, 2018 at 8:04

The typical shift that happens when moving to fields which aren't algebraically closed (or more general rings) is that one redefines what $$\Bbb A^n_R$$ and $$\Bbb P^n_R$$ are for the the ring $$R$$. Instead of declaring that $$\Bbb A^n_R$$ is the set $$R^n$$ with a funny topology, one says that $$\Bbb A^n_R = \operatorname{Spec} R[x_1,\cdots,x_n]$$, and similarly, instead of regarding $$\Bbb P^n_R$$ as "lines through the origin in $$R^{n+1}$$", we think of it instead as lines through the origin in $$\Bbb A^{n+1}_R = \operatorname{Spec} R[x_0,\cdots,x_n]$$ (also known as $$\operatorname{Proj} R[x_0,\cdots,x_n]$$). Sometimes you'll also see people drop the "irreducible" adjective - the general formulation is that an affine (resp. projective) variety is a closed subscheme of $$\Bbb A^n$$ (resp. $$\Bbb P^n$$) satisfying some adjectives, which often vary from author to author.
This redefinition gives you many more points in $$\Bbb A^n_K$$ if $$K$$ is not an algebraically closed field - for any $$K'$$ a Galois extension of $$K$$, we get a point for every Galois orbit in $$K'$$. Having access to these points means the sorts of shenanigans you're attempting to pull off in the linked question don't work - every polynomial's zero set has the correct dimension, etc.