Abstract : Calling a field K almost arbitrary (a.a.) if it admits finite extensions of degree >2 and prime to the characteristic of K we prove the following
Theorem:
Any perfect a.a. field K is up to isomorphism encoded in the absolute Galois group of the rational function field K(t) over K.
We will also present the "local theory" of birational anabelian geometry characterizing K-rational points on arbitrary curves over an a.a. perfect field K in Galois theoretic terms, and give applications of the above Theorem to questions of decidability.