Galois Points on Varieties

A field K is ample if for every geometrically integral K-variety V with a smooth K-point, V(K) is Zariski dense in V. A field K is Galois-potent if every geometrically integral K-variety has a closed point whose residue field is Galois over K. We prove that every ample field is Galois-potent. But we construct also non-ample Galois-potent fields; in fact, every field has a regular extension with these properties.