closure of a set of k-rational points always defined over k?

Question

Is it correct to say that the Zariski closure of any set of $k$-rational points in affine space over $\overline{k}$ is always defined over $k$?

Answer 1

While the question can be answered briefly and narrowly in an ad hoc style (as Peter Mueller has done), it may be useful to add a reference to Borel's textbook while putting the question in context.

The language of $k$-closed sets and $k$-varieties (varieties defined over a field $k$) was developed during the evolution of modern algebraic geometry primarily to deal with fields of prime characteristic. But eventually the more flexible language of schemes over arbitrary commutative rings supplanted some of this language, at the same time explaining better what the underlying difficulty was. In his lectures on linear algebraic groups at Columbia in the late 1960s, Borel tried to incorporate some of the scheme viewpoint while keeping the story short. This material was written up by Bass in the preliminary Chapter AG of the W.A. Benjamin edition published in 1969. Eventually Borel expanded the coverage in a second edition, published in 1991 as Springer GTM 126. But the earlier material was essentially unchanged.

Sections 12-14 of Chapter AG are especially relevant here. It is shown that a $k$-closed subset (say of an affine variety over $k$) is always defined over a purely inseparable extension of $k$. This focuses attention on the case of imperfect fields in charactristic $p>0$. In his $\S14$, Borel develops a Galois criterion using the Galois group of a separable closure of $k$ to decide which $k$-closed sets are in fact defined over $k$. His Corollary 14.6 contains the particular result asked about here: take his $Z_i$ to be sets consisting of single points over $k$ and $Z$ to be the closure of their union.

It may or may not be statisfactory in many areas of algebraic geometry to deal in this way with varieties over fields rather than schemes over rings, but the language Borel adopted does allow one to develop more rapidly the basic ideas about structure of semisimple or reductive groups over fields. In the longer run, the modern language used in the recent book Pseudo-reductive Groups by Conrad-Gabber-Prasad is necessary for applications to number theory, etc.

Answer 2

Yes, that's the case. Let $f\in\bar k[\mathbf x]$ be a polynomial vanishing on your set $M$ of $k$-rational points. Let $E$ be the field generated by $k$ and the coefficients of $f$, and let $\alpha_1,\dots,\alpha_n$ be a basis of the $k$-space $E$. Write $f=\sum\alpha_i f_i$ with $f_i\in k[\mathbf x]$. Then $f(m)=0$ for $m\in M$ and $f_i(m)\in k$, implying $f_i(m)=0$ for all $i$. So $f$ and $\{f_1,f_2\dots,f_n\}$ describe the same algebraic set. So you can replace any polynomial with coefficients not in $k$ by polynomials from $k[\mathbf x]$.