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.