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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!

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    $\begingroup$ 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 $(x^2+1)$. $\endgroup$
    – J1U
    Oct 7, 2018 at 7:49
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    $\begingroup$ 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 $k^n \subset \bar{k}^n$. $\endgroup$
    – J1U
    Oct 7, 2018 at 8:04

1 Answer 1

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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.

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