Tchebotarev theorems for function fields

The central theme of the paper is the specialization of algebraic function field extensions. Our main results are Tchebotarev type theorems for Galois function field extensions, finite or infinite, over various base fields: under some conditions, we extend the classical finite field case to number fields, p-adic fields, PAC fields, function fields κ(x), etc. We also compare the Tchebotarev conclusion - existence of unramified local specializations with Galois group any cyclic subgroup of the generic Galois group (up to conjugation) - to the Hilbert specialization property. For a function field extension with the Tchebotarev property, the exponent of the Galois group is bounded by the l.c.m. of the local specialization degrees. Local-global questions arise for which we provide answers, examples and counter-examples.