  We prove Tchebotarev type theorems for function field extensions over various
base fields: number fields, finite fields, p-adic fields, PAC fields, etc. The
Tchebotarev conclusion - existence of appropriate cyclic residue extensions -
also compares to the Hilbert specialization property. It is more local but
holds in more situations and extends to infinite extensions. For a function
field extension satisfying the Tchebotarev conclusion, the exponent of the
Galois group is bounded by the l.c.m. of the local specialization degrees.
Further local-global questions arise for which we provide answers, examples and
counter-examples.
