Algebraic geometry over general fields via Galois theory

Question

I was informed by my supervisor that it is possible to study algebraic sets over non-algebraically closed fields by passing to an algebraically closed setting via Galois theory. How exactly do we do this?

Answer 1

Let $k$ be a (perfect) field, $\bar{k}$ a fixed algebraic closure, and let $G_{\bar{k}/k}$ be the Galois group of the extension $\bar{k}/k$. You may define affine and projective spaces as usual, i.e., over $\bar{k}$. However, it makes sense to let $$\mathbb A^n(k) = \left\{(x_1,...,x_n):x_i \in k\right\} \subset \mathbb A^n(\bar{k}),$$ and similarly $\mathbb P^n(k) \subset \mathbb P^n(\bar{k})$. In terms of Galois theory, we note that $$\mathbb A^n(k) = \left\{p \in \mathbb A^n(\bar{k}): p^\sigma = p~\mathrm{for~all}~\sigma \in G_{\bar{k}/k}\right\},$$ where $p = (x_1,...,x_n)$ and $p^\sigma = (x_1^\sigma,...,x_n^{\sigma})$, $\sigma \in G$. Similarly, if $f \in k[x_1,...,x_n]$ and $p \in \mathbb A^n$, then $$f(p^\sigma) = f(p)^\sigma.$$ Thus if we have an algebraic set $V$ over $\bar{k}$ in the usual sense, then we can define $$V(k) = \left\{p \in V:p^\sigma = p~\mathrm{for~all}~\sigma \in G\right\}.$$ There is much more you could generalize, but I'll stop here. If you are interested, please read Silverman's The Arithmetic of Elliptic Curves.