The p-adic analog of Artin-Schreier Theorem – revisited (II)

A famous Theorem by Artin and Schreier characterizes the real closed fields as being those fields which have a finite non-trivial absolute Galois group. Instances of p-adic analogs of this Theorem are known (Neukirch, Pop, Koenigsmann, Efrat), but there is much more to this story. Namely I will give a ‘minimalistic’ p-adic analog, which as in the Artin-Schreier Theorem, invoves only finite groups. This aspect of the story relates to the birational p-adic section conjecture, etc.