-Adic Dirichlet characters of algebraic function fields

A theory of precyclic P-extensions is developed which contains the well-known Witt theory. This theory makes it possible to describe so-called P-adic Dirichlet characters of function fields. In particular, for a trigonometric sum of the form[Figure not available: see fulltext.] where P is a prime number and f is a polynomial, its expression in terms of the zeros of a certain L function of the base field is obtained