Action of The Absolute Galois Group
The absolute Galois group Gal(kalg/k) of k naturally acts on the zero-locus in An(kalg) of a subset of the polynomial ring k. In general, if V is a scheme over k (e.g. a k-algebraic set), Gal(kalg/k) naturally acts on V ×Spec(k) Spec(kalg) via its action on Spec(kalg).
When V is a variety defined over a perfect field k, the scheme V can be recovered from the scheme V ×Spec(k) Spec(kalg) together with the action of Gal(kalg/k) on the latter scheme: the sections of the structure sheaf of V on an open subset U are exactly the sections of the structure sheaf of V ×Spec(k) Spec(kalg) on U ×Spec(k) Spec(kalg) whose residues are constant on each Gal(kalg/k)-orbit in U ×Spec(k) Spec(kalg). In the affine case, this means the action of the absolute Galois group on the zero-locus is sufficient to recover the subset of k consisting of vanishing polynomials.
In general, this information is not sufficient to recover V. In the example of the zero-locus of x1p- t in (Fp(t))alg, the variety consists of a single point and so the action of the absolute Galois group cannot distinguish whether the ideal of vanishing polynomials was generated by x1 - t1/p, by x1p- t, or, indeed, by x1 - t1/p raised to some other power of p.
For any subfield L of kalg and any L-variety V, an automorphism σ of kalg will map V isomorphically onto a σ(L)-variety.
Read more about this topic: Field Of Definition
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