Anabelian Phenomena in Geometry and Arithmetic

a) Absolute Galois group and real fields LetK be an arbitrary field,Ka an algebraic extension,Ks the separable extension ofK insideKa, and finally GK = Aut(Ks|K) = Aut(Ka|K) the absolute Galois group of K. It is a celebrated well known Theorem by Artin–Schreier from the 1920’s which asserts the following: If GK is a finite non-trivial group, then GK ∼= GR and K is real closed. In particular, char(K) = 0, and Ka = K[ √ −1]. Thus the non-triviality + finiteness of GK imposes very strong restrictions on K. Nevertheless, the kind of restrictions imposed on K are not on the isomorphism type of K as a field, as there is a big variety of isomorphy types of real closed fields (and their classification up to isomorphism seems to be out of reach). The kind of restriction imposed on K is rather one concerning the algebraic behavior of K, namely that the algebraic geometry over K looks like the one over R.